1

FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 4 Chapter 4 Kinematics of Fluid Motion Kinematics of Fluid Motion

JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering

National Taiwan UniversityNational Taiwan University0928200909282009

2

MAIN TOPICSMAIN TOPICS

The Velocity FieldThe Velocity FieldThe Acceleration FieldThe Acceleration FieldControl Volume and System RepresentationControl Volume and System RepresentationThe Reynolds Transport TheoremThe Reynolds Transport Theorem

3

Field Representation of flow Field Representation of flow 1212

At a given instant in time At a given instant in time any fluid propertyany fluid property (such (such as density pressure velocity and acceleration) as density pressure velocity and acceleration) can can be described as a functions of thebe described as a functions of the fluidfluidrsquorsquos locations locationThis representation of fluid parameters as This representation of fluid parameters as functions of the spatial coordinates is termed a functions of the spatial coordinates is termed a field representation of flowfield representation of flow何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度表達成空間座標的函數稱為流體的場表徵表達成空間座標的函數稱為流體的場表徵

當當flowflow內的任何參數均如此表達則該內的任何參數均如此表達則該flowflow稱為稱為flow fieldflow field

4

Field Representation of flow Field Representation of flow 2222

The specific The specific field representationfield representation may be different at may be different at different times so that to describe a fluid flow we must different times so that to describe a fluid flow we must determine the various parameter determine the various parameter not only as functions of not only as functions of the spatial coordinates but also as a function of timethe spatial coordinates but also as a function of time

EXAMPLE Temperature field EXAMPLE Temperature field T = T ( x y z t )T = T ( x y z t )EXAMPLE Velocity field EXAMPLE Velocity field

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空間關係外也必須考慮時間因素間關係外也必須考慮時間因素

溫度場與速度場內的溫度速度可表達成時間與空間函數溫度場與速度場內的溫度速度可表達成時間與空間函數

5

Velocity FieldVelocity Field 1212

The velocity at any particle in the flow field (the velocity The velocity at any particle in the flow field (the velocity field) is given byfield) is given by

where u v and w are the x y and z components of the where u v and w are the x y and z components of the velocity vectorvelocity vector

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

)tzyx(VV =

在在velocity fieldvelocity field中任何質點的速度（參數之一）均可寫成時間與空間的函數中任何質點的速度（參數之一）均可寫成時間與空間的函數

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

2

MAIN TOPICSMAIN TOPICS

The Velocity FieldThe Velocity FieldThe Acceleration FieldThe Acceleration FieldControl Volume and System RepresentationControl Volume and System RepresentationThe Reynolds Transport TheoremThe Reynolds Transport Theorem

3

Field Representation of flow Field Representation of flow 1212

At a given instant in time At a given instant in time any fluid propertyany fluid property (such (such as density pressure velocity and acceleration) as density pressure velocity and acceleration) can can be described as a functions of thebe described as a functions of the fluidfluidrsquorsquos locations locationThis representation of fluid parameters as This representation of fluid parameters as functions of the spatial coordinates is termed a functions of the spatial coordinates is termed a field representation of flowfield representation of flow何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度表達成空間座標的函數稱為流體的場表徵表達成空間座標的函數稱為流體的場表徵

當當flowflow內的任何參數均如此表達則該內的任何參數均如此表達則該flowflow稱為稱為flow fieldflow field

4

Field Representation of flow Field Representation of flow 2222

The specific The specific field representationfield representation may be different at may be different at different times so that to describe a fluid flow we must different times so that to describe a fluid flow we must determine the various parameter determine the various parameter not only as functions of not only as functions of the spatial coordinates but also as a function of timethe spatial coordinates but also as a function of time

EXAMPLE Temperature field EXAMPLE Temperature field T = T ( x y z t )T = T ( x y z t )EXAMPLE Velocity field EXAMPLE Velocity field

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空間關係外也必須考慮時間因素間關係外也必須考慮時間因素

溫度場與速度場內的溫度速度可表達成時間與空間函數溫度場與速度場內的溫度速度可表達成時間與空間函數

5

Velocity FieldVelocity Field 1212

The velocity at any particle in the flow field (the velocity The velocity at any particle in the flow field (the velocity field) is given byfield) is given by

where u v and w are the x y and z components of the where u v and w are the x y and z components of the velocity vectorvelocity vector

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

)tzyx(VV =

在在velocity fieldvelocity field中任何質點的速度（參數之一）均可寫成時間與空間的函數中任何質點的速度（參數之一）均可寫成時間與空間的函數

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

3

Field Representation of flow Field Representation of flow 1212

At a given instant in time At a given instant in time any fluid propertyany fluid property (such (such as density pressure velocity and acceleration) as density pressure velocity and acceleration) can can be described as a functions of thebe described as a functions of the fluidfluidrsquorsquos locations locationThis representation of fluid parameters as This representation of fluid parameters as functions of the spatial coordinates is termed a functions of the spatial coordinates is termed a field representation of flowfield representation of flow何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度何謂的流體的「場表徵」將流體的特性如密度壓力速度與加速度表達成空間座標的函數稱為流體的場表徵表達成空間座標的函數稱為流體的場表徵

當當flowflow內的任何參數均如此表達則該內的任何參數均如此表達則該flowflow稱為稱為flow fieldflow field

4

Field Representation of flow Field Representation of flow 2222

The specific The specific field representationfield representation may be different at may be different at different times so that to describe a fluid flow we must different times so that to describe a fluid flow we must determine the various parameter determine the various parameter not only as functions of not only as functions of the spatial coordinates but also as a function of timethe spatial coordinates but also as a function of time

EXAMPLE Temperature field EXAMPLE Temperature field T = T ( x y z t )T = T ( x y z t )EXAMPLE Velocity field EXAMPLE Velocity field

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空間關係外也必須考慮時間因素間關係外也必須考慮時間因素

溫度場與速度場內的溫度速度可表達成時間與空間函數溫度場與速度場內的溫度速度可表達成時間與空間函數

5

Velocity FieldVelocity Field 1212

The velocity at any particle in the flow field (the velocity The velocity at any particle in the flow field (the velocity field) is given byfield) is given by

where u v and w are the x y and z components of the where u v and w are the x y and z components of the velocity vectorvelocity vector

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

)tzyx(VV =

在在velocity fieldvelocity field中任何質點的速度（參數之一）均可寫成時間與空間的函數中任何質點的速度（參數之一）均可寫成時間與空間的函數

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

4

Field Representation of flow Field Representation of flow 2222

The specific The specific field representationfield representation may be different at may be different at different times so that to describe a fluid flow we must different times so that to describe a fluid flow we must determine the various parameter determine the various parameter not only as functions of not only as functions of the spatial coordinates but also as a function of timethe spatial coordinates but also as a function of time

EXAMPLE Temperature field EXAMPLE Temperature field T = T ( x y z t )T = T ( x y z t )EXAMPLE Velocity field EXAMPLE Velocity field

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空由於「場表徵」會因時間而異因此在描述流體參數時除了考慮其空間關係外也必須考慮時間因素間關係外也必須考慮時間因素

溫度場與速度場內的溫度速度可表達成時間與空間函數溫度場與速度場內的溫度速度可表達成時間與空間函數

5

Velocity FieldVelocity Field 1212

The velocity at any particle in the flow field (the velocity The velocity at any particle in the flow field (the velocity field) is given byfield) is given by

where u v and w are the x y and z components of the where u v and w are the x y and z components of the velocity vectorvelocity vector

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

)tzyx(VV =

在在velocity fieldvelocity field中任何質點的速度（參數之一）均可寫成時間與空間的函數中任何質點的速度（參數之一）均可寫成時間與空間的函數

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

5

Velocity FieldVelocity Field 1212

The velocity at any particle in the flow field (the velocity The velocity at any particle in the flow field (the velocity field) is given byfield) is given by

where u v and w are the x y and z components of the where u v and w are the x y and z components of the velocity vectorvelocity vector

k)tzyx(wj)tzyx(vi)tzyx(uV ++=

)tzyx(VV =

在在velocity fieldvelocity field中任何質點的速度（參數之一）均可寫成時間與空間的函數中任何質點的速度（參數之一）均可寫成時間與空間的函數

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

6

Velocity FieldVelocity Field 2222

The velocity of a particle is the time rate The velocity of a particle is the time rate of change of the position vector for that of change of the position vector for that particleparticle

dtrdV A

A

rr= 質點的速度為其位置質點的速度為其位置

向量的時間改變率向量的時間改變率

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

7

Example 41 Velocity Field RepresentationExample 41 Velocity Field Representation

A velocity field is given by whereA velocity field is given by where VV00 and and ll are are constants constants At what location in the flow field is the speed equal to At what location in the flow field is the speed equal to VV00Make a sketch of the velocity fieldMake a sketch of the velocity field in the first quadrant (xin the first quadrant (x≧≧0 y 0 y ≧≧0) 0) by drawing arrows representing the fluid velocity at representatby drawing arrows representing the fluid velocity at representative ive locationslocations

)jyix)(V(V 0

rrl

rminus=

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

8

Example 41 Example 41 SolutionSolution

2122021222 )yx(V)wvu(V +=++=l

The x y and z components of the velocity are given by u = VThe x y and z components of the velocity are given by u = V00xxll v v = = --VV00y y ll and w = 0 so that the fluid speed V and w = 0 so that the fluid speed V

The speed is V = VThe speed is V = V00 at any location on the circle of radius at any location on the circle of radius llcentered at the origin [(xcentered at the origin [(x22 + y+ y22))1212= = ll] as shown in Figure E41 (a) ] as shown in Figure E41 (a)

The direction of the fluid velocity relative to the x axis is giThe direction of the fluid velocity relative to the x axis is given in ven in terms of terms of θθ= = arctan(vuarctan(vu)) as shown in Figure E41 (b) For this flow as shown in Figure E41 (b) For this flow

xy

xVyV

uvθtan

0

0 minus=

minus==

l

l

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

9

About flowing fluidAbout flowing fluidhelliphellip

Method of DescriptionMethod of DescriptionSteady and Unsteady FlowsSteady and Unsteady Flows1D 2D and 3D Flows1D 2D and 3D FlowsTimelines Timelines PathlinesPathlines StreaklinesStreaklines and Streamlines and Streamlines

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

10

Methods of DescriptionMethods of Description

Lagragian method = System methodEulerian method = Control volume method

在在velocity fieldvelocity field描述流體參數的方法描述流體參數的方法

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

11

LagragianLagragian MethodMethod

Following individual fluid particles as they moveFollowing individual fluid particles as they moveThe fluid particles are tagged or identifiedThe fluid particles are tagged or identifiedDetermining how the fluid properties associated with these Determining how the fluid properties associated with these particles change as a function of timeparticles change as a function of timeExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular fluid particle A and record that particleparticular fluid particle A and record that particlersquorsquos temperature as it s temperature as it moves about moves about TTAA = T= TAA (t)(t) The use of may such measuring devices The use of may such measuring devices moving with various fluid particles would provide the temperaturmoving with various fluid particles would provide the temperature e of these fluid particles as a function of timeof these fluid particles as a function of time

個別質點上貼標籤跟著它觀察它鑑（界）定它個別質點上貼標籤跟著它觀察它鑑（界）定它

觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測觀察紀錄個別質點的特性與時間的關係例如在個別質點貼上溫度量測儀器偵測與紀錄質點移動過程中溫度與時間的關係位置儀器偵測與紀錄質點移動過程中溫度與時間的關係位置因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪因此若要把時間與空間的關係弄清楚還需要去知道「觀察者」在哪

一種以動制動的方法一種以動制動的方法

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

12

Following the ParticlesFollowing the Particles

ExperimentExperiment

ComputerComputer

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

13

EulerianEulerian MethodMethod

Use the field conceptUse the field conceptThe fluid motion is given by completely prescribing the The fluid motion is given by completely prescribing the necessary properties as a functions of space and timenecessary properties as a functions of space and timeObtaining information about the flow in terms of what Obtaining information about the flow in terms of what happens at fixed points in space as the fluid flows past happens at fixed points in space as the fluid flows past those pointsthose pointsExampleExample one attaches the temperatureone attaches the temperature--measuring device to a measuring device to a particular point (xyz) and record the temperature at that poinparticular point (xyz) and record the temperature at that point as a t as a function of time function of time T = T ( x y z t )T = T ( x y z t )

具有「具有「fieldfield」的觀察在一已知固定點觀察經過該點的流體」的觀察在一已知固定點觀察經過該點的流體

一種以一種以靜靜制動的方法制動的方法

因為觀察者的位置已知只要記錄時間的關係即可清楚交代因為觀察者的位置已知只要記錄時間的關係即可清楚交代propertiesproperties的時間與空間關係的時間與空間關係

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

14

1D 2D and 3D Flows1D 2D and 3D Flows

Depending on Depending on the number of space coordinates required to the number of space coordinates required to specify the flow fieldspecify the flow fieldAlthough most flow fields are inherently threeAlthough most flow fields are inherently three--dimensional analysis based on dimensional analysis based on fewer dimensions is fewer dimensions is frequently meaningfulfrequently meaningfulThe complexity of analysis increases considerably with The complexity of analysis increases considerably with the number of dimensions of the flow fieldthe number of dimensions of the flow field

需要多少維度才可以清楚描述流體的需要多少維度才可以清楚描述流體的參數（特性）當然維度越多分參數（特性）當然維度越多分析問題的複雜度與難度就越高析問題的複雜度與難度就越高

Flow fieldFlow field內大多數的流體特性（參數）本質上均需要透過內大多數的流體特性（參數）本質上均需要透過三個維度才能描述清楚為簡化問題難度可在不影響結果三個維度才能描述清楚為簡化問題難度可在不影響結果精度的前提下忽略其中的一或二個維度精度的前提下忽略其中的一或二個維度

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

15

33--D flow visualizationD flow visualization

Flow past a wingFlow past a wing

Flow visualization of the Flow visualization of the complex threecomplex three--dimensional flow dimensional flow past a model wingpast a model wing

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

16

Steady and Unsteady Flows Steady and Unsteady Flows 1212

Steady flow the properties at every point in a flow Steady flow the properties at every point in a flow field field do not change with timedo not change with time

where where ηη represents any fluid propertyrepresents any fluid propertyUnsteady flowUnsteady flowhelliphellip Change with timeChange with time

NonperiodicNonperiodic flow periodic flow and truly random flow periodic flow and truly random flowflowMore difficult to analyzeMore difficult to analyze

0t=

partηpart 任何任何flow fieldflow field內的內的propertyproperty

該該propertyproperty與時間的變化無關與時間的變化無關

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

17

Steady and Unsteady Flows Steady and Unsteady Flows 2222

Steady or unsteady Steady or unsteady Observed at a fixed point in Observed at a fixed point in spacespaceFor steady flow the values of all fluid properties at For steady flow the values of all fluid properties at any fixed point are independent of time However any fixed point are independent of time However the value of these properties for a given fluid the value of these properties for a given fluid particle may change with time as the particle flowsparticle may change with time as the particle flows

SteadySteady或或unsteadyunsteady是從固定點加以觀察當質點移動後即使是從固定點加以觀察當質點移動後即使是是steady flowsteady flow該質點特性的該質點特性的rdquordquovaluevaluerdquordquo也會改變也會改變

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

18

Path of Fluid ParticlePath of Fluid Particle

TimelineTimelinePathlinePathlineStreaklineStreaklineStreamlineStreamline

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

19

Streamlines Streamlines 1212

Streamline Line drawn in the flow field so that at a given Streamline Line drawn in the flow field so that at a given instant they are tangent to the direction of flow at every instant they are tangent to the direction of flow at every point in the flow field gtgtgt No flow across a streamlinepoint in the flow field gtgtgt No flow across a streamline

Streamline is everywhere tangent to the velocity fieldStreamline is everywhere tangent to the velocity fieldIf the flow is steady nothing at a fixed point changes If the flow is steady nothing at a fixed point changes with time so the streamlines are fixed lines in spacewith time so the streamlines are fixed lines in spaceFor unsteady flows the streamlines may change shape For unsteady flows the streamlines may change shape with timewith timeStreamlines are obtained analytically by integrating the Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity fieldequations defining lines tangent to the velocity field

實驗室實驗室不不可能體驗的線可能體驗的線

虛幻的在虛幻的在flow fieldflow field中與質點速度相切的線稱為中與質點速度相切的線稱為streamlinestreamline

由由Velocity fieldVelocity field積分得到積分得到StreamlineStreamline的解析解的解析解

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

20

Streamlines Streamlines 22 22

For two dimensional flows the slope of the streamline For two dimensional flows the slope of the streamline dydxdydx must be equal to the tangent of the angle that the must be equal to the tangent of the angle that the velocity vector makes with the x axisvelocity vector makes with the x axis

Steady flow Steady flow pathlinespathlines streaklinesstreaklines and streamlines are identical lines and streamlines are identical linesUnsteady flow Unsteady flow pathlinespathlines streaklinesstreaklines and streamlines are not coincide and streamlines are not coincide

uv

dxdy

= If the velocity field is known If the velocity field is known as a function of x and y this as a function of x and y this equation can be integrated to equation can be integrated to give the equation of give the equation of streamlinesstreamlines

StreamlineStreamline的斜率與速度向量的關係的斜率與速度向量的關係

速度向量與速度向量與XX軸的夾角軸的夾角四種四種LinesLines的關係的關係

如果如果particleparticle的速度的速度分量已知分量已知此式積分此式積分即可求出即可求出StreamlineStreamline

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

21

StreaklinesStreaklines

StreaklineStreakline Line jointing the fluid particles passing Line jointing the fluid particles passing through one fixed location in space through one fixed location in space StreaklinesStreaklines are more are more of a laboratory tool than an analytical toolof a laboratory tool than an analytical tool

StreaklinesStreaklines can be obtained by taking instantaneous can be obtained by taking instantaneous photographysphotographys of of makedmaked particles that all passed particles that all passed through a given location in the flow field at some through a given location in the flow field at some earlier timeearlier timeUsing dye or smoke at a fixed location in space to Using dye or smoke at a fixed location in space to identify all fluid particles pass through this pointidentify all fluid particles pass through this point

將通過特定位置的質點串連起來的線稱為將通過特定位置的質點串連起來的線稱為 StreaklineStreakline

在特定位置上注入染料或煙來在特定位置上注入染料或煙來註記流經過該位置的質點註記流經過該位置的質點之後選一個時間拍照記錄這些之後選一個時間拍照記錄這些質點串連起來的線即可得質點串連起來的線即可得到到 StreaklineStreakline

實驗室可以體驗的線實驗室可以體驗的線

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

22

PathlinesPathlines

PathlinePathline Path or trajectory traced out by Path or trajectory traced out by a moving fluid a moving fluid particleparticle

Using dye or smoke to identify a fluid particle at a Using dye or smoke to identify a fluid particle at a given instant and then take given instant and then take a long exposure a long exposure photographphotograph of its subsequent motion The line traced of its subsequent motion The line traced out by the particle is a out by the particle is a pathlinepathline

在特定時間利用染料或煙來註記一個流體質在特定時間利用染料或煙來註記一個流體質點之後透過長期曝光記錄該質點在曝光時間內點之後透過長期曝光記錄該質點在曝光時間內跑過的路徑該路經稱之為跑過的路徑該路經稱之為 PathlinePathline

實驗室可以體驗的線實驗室可以體驗的線

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

23

Example 42 Streamlines for a Given Example 42 Streamlines for a Given Velocity FieldVelocity Field

Determine the Determine the streamlinesstreamlines for the twofor the two--dimensional steady dimensional steady flow discussed in Example 41 flow discussed in Example 41 )jyix)(V(V 0

rrl

rminus=

Figure E42

ParticleParticle的速度分量的速度分量

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

24

Example 42 Example 42 SolutionSolution

xy

x)V(y)V(

uv

dxdy

0

0 minus=minus

==l

l

y)V(vandx)V(u 00 ll minus==SinceSince

The streamlines are given by solution of the equationThe streamlines are given by solution of the equation

IntegratingIntegratinghelliphellip

ttanconsxlnylnorx

dxy

dy+minus=minus= intint

The streamline is The streamline is xyxy = C= C where C is a constant where C is a constant

積分即可求出積分即可求出StreamlineStreamline

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

25

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines1212

Water flowing from the oscillating slit shown in Figure E43a Water flowing from the oscillating slit shown in Figure E43a produces a velocity field given by produces a velocity field given by VV=u=u00sin[sin[ωω(t(t--yvyv00))]]ii++vv00jj where where uu00 v v00 and and ωω are constantsare constants Thus the y component of velocity Thus the y component of velocity remains constant (v=vremains constant (v=v00) and the x component of velocity at y=0 ) and the x component of velocity at y=0 coincides with the velocity of the oscillating sprinkler head coincides with the velocity of the oscillating sprinkler head [u=u[u=u00sin(sin(ωωt) at y=0]t) at y=0] (a) Determine the streamline that passes (a) Determine the streamline that passes through the origin at t=0 at t=through the origin at t=0 at t=ππ22ωω (b) Determine the (b) Determine the pathlinepathline of of the particle that was at the origin at t=0 at the particle that was at the origin at t=0 at tt ==ππ22 (c) Discuss the (c) Discuss the shape of the shape of the streaklinestreakline that passes through the originthat passes through the origin

ParticleParticle的速度分量的速度分量

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

26

Example 43 Comparison of Streamlines Example 43 Comparison of Streamlines PathlinesPathlines and Streaklines and Streaklines2222

Figure E43

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

27

Example 43 Example 43 SolutionSolution1818

intint =⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusω dxvdy

vytsinu 00

0

( )[ ]00

0

vytsinuv

uv

dxdy

minusω==

Since u=uSince u=u00sin[sin[ωω(t(t--yvyv00))] and v=v] and v=v00 the streamlines are given by the streamlines are given by the solution ofthe solution of

IntegratingIntegratinghelliphellip

( ) Cxvvytcosvu 00

00 +=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minusωω (1)(1)

(a)(a)

where where CC is a constantis a constant

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

28

Example 43 Example 43 SolutionSolution2828

⎟⎟⎠

⎞⎜⎜⎝

⎛minus=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛minus=

0

0

0

0

2cos

2cos

vyu

vyux ωπ

ωωπω

ω

⎥⎦

⎤⎢⎣

⎡minus⎟⎟

⎠

⎞⎜⎜⎝

⎛ ωω

= 1v

ycosux0

0

For the streamline at t=0 that passes through the origin (x=y=0)For the streamline at t=0 that passes through the origin (x=y=0) EqEq 1 gives the value of 1 gives the value of CC=u=u00vv00ωω The equation for this streamline is The equation for this streamline is

Similarly for the streamline at t=Similarly for the streamline at t=ππ22ωω that passes through the that passes through the origin origin EqEq 1 gives 1 gives CC=0 The equation for this streamline=0 The equation for this streamline

(2)(2)

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωω

=0

0

vysinux (3)(3)

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

29

Example 43 Example 43 SolutionSolution3838

⎥⎦

⎤⎢⎣

⎡minusω= )

vyt(sinu

dtdx

00 0v

dtdy

=

The The pathlinepathline of a particle can be obtained from the velocity field and of a particle can be obtained from the velocity field and definition of the velocity definition of the velocity Since u=Since u=dxdtdxdt and v=and v=dydtdydt

andand

Integrated to give the y coordinate of the Integrated to give the y coordinate of the pathlinepathline

10 Ctvy += (4)(4)

(b)(b)

Where Where CC11 is a constant is a constant

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

30

Example 43 Example 43 SolutionSolution4848

⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=⎥

⎦

⎤⎢⎣

⎡ +minusω=

0

10

0

100 v

Csinu)v

Ctvt(sinudtdx

20

10 Ct

vCsinux +⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ωminus=

tvy 0=

Integrated to give the x component of the Integrated to give the x component of the pathlinepathline

where where CC22 is a constant is a constant

andand0=x (6)(6)

(5)(5)

With this known y=With this known y=y(ty(t) dependence the x equation for the ) dependence the x equation for the pathlinepathlinebecomesbecomes

For the particle that was at the origin (x=y=0) at time t=0 For the particle that was at the origin (x=y=0) at time t=0 EqsEqs 4 4 and 5 give and 5 give CC11==CC22=0 Thus the =0 Thus the pathlinepathline isis

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

31

Example 43 Example 43 SolutionSolution5858

⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tux 0 ⎟⎠⎞

⎜⎝⎛

ωπ

minus=2

tvy 0

Similarly for the particle that was at the origin at t=Similarly for the particle that was at the origin at t=ππ22ωω EqsEqs 4 4 and 5 give and 5 give CC11==--ππvv0022ωωand and CC22==--ππuu0022ωω Thus the Thus the pathlinepathline for for this particle isthis particle is

The The pathlinepathline can be drawn by plotting the locus of can be drawn by plotting the locus of x(tx(t) ) y(ty(t) value ) value for tfor tgege0 or by eliminating the parameter t from Eq7 to give0 or by eliminating the parameter t from Eq7 to give

xuvy

0

0=

andand

(8)(8)

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

32

Example 43 Example 43 SolutionSolution6868

The The pathlinespathlines given by given by EqsEqs 6 and 8 shown in Figure E43c 6 and 8 shown in Figure E43c are straight lines from the origin (rays) The are straight lines from the origin (rays) The pathlinespathlines and and streamlines do not coincide because the flow is unsteadystreamlines do not coincide because the flow is unsteady

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

33

Example 43 Example 43 SolutionSolution7878

The The streaklinestreakline through the origin at time t=0 is the locus of through the origin at time t=0 is the locus of particles at t=0 that previously (tlt0) passed through the origiparticles at t=0 that previously (tlt0) passed through the origin n The general shape of the The general shape of the streaklinesstreaklines can be seen as followscan be seen as followsEach particle that flows through the origin travels in a straighEach particle that flows through the origin travels in a straight line t line ((pathlinespathlines are rays from the origin) the slope of which lies are rays from the origin) the slope of which lies between between plusmnplusmnvv00uu00 as shown in Figure E43d Particles passing as shown in Figure E43d Particles passing through the origin at different times are located on different rthrough the origin at different times are located on different rays ays from the origin and at different distances from the originfrom the origin and at different distances from the origin

(c)(c)

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

34

Example 43 Example 43 SolutionSolution8888

The net result is that a stream of dye continually injected at tThe net result is that a stream of dye continually injected at the he origin (a origin (a streaklinestreakline) would have the shape shown in Figure ) would have the shape shown in Figure E43d Because of the unsteadiness the E43d Because of the unsteadiness the streaklinestreakline will vary will vary with time although it will always have the oscillating sinuouswith time although it will always have the oscillating sinuouscharacter shown Similar character shown Similar streaklinesstreaklines are given by the stream of are given by the stream of water from a garden hose nozzle that oscillates back and forth iwater from a garden hose nozzle that oscillates back and forth in n a direction normal to the axis of the nozzle In this example a direction normal to the axis of the nozzle In this example neither the streamlines neither the streamlines pathlinespathlines nor nor streaklinesstreaklines coincide If the coincide If the flow were steady all of these lines would be the sameflow were steady all of these lines would be the same

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

35

Description ofDescription of AccelerationAcceleration

For For LagrangianLagrangian methodmethod the fluid acceleration is described the fluid acceleration is described as done in solid body dynamicsas done in solid body dynamics

For For EulerianEulerian methodmethod the fluid acceleration is described as the fluid acceleration is described as function of position and time without actually following function of position and time without actually following any particlesany particles

)t(aarr

=

)tzyx(aarr

=

加速加速度是時間函數度是時間函數

加速加速度是時間與空間函數度是時間與空間函數

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

36

Acceleration FieldAcceleration Field－－Material DerivativeMaterial Derivative 1515

The acceleration of a fluid particle for use in NewtonThe acceleration of a fluid particle for use in Newtonrsquorsquos s second law issecond law is

This is incorrect because is a field ie it describesThis is incorrect because is a field ie it describesthe whole flow and not just the motion of an individual the whole flow and not just the motion of an individual particleparticleThe problem is Given the velocity fieldThe problem is Given the velocity field

find the acceleration of a fluid particlefind the acceleration of a fluid particle

dtVdarr

= YES or NO YES or NO

Vr

)tzyx(VVrr

=

從從fieldfield角度速度不只是時間的函數角度速度不只是時間的函數

速度是時間與空間的函數速度是時間與空間的函數

個別質點的速度與加速度個別質點的速度與加速度關係關係

從從fieldfield角度速度與加角度速度與加速度關係並非如此速度關係並非如此

依此概念如何延伸定義加速度場依此概念如何延伸定義加速度場

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

37

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 2525

The velocity of a fluid particle A in space at time t

The velocity of a fluid particle in space at time t+dt)tzyx(VV AtA

rr=

)dttdzzdyydxx(VV AdttA ++++=+

rr

rdr rr+rr

The change in the velocity of the particle in moving from location to is given by the chain rule

dtt

Vdzz

Vdyy

Vdxx

VVd AA

AA

AA

AA part

part+

partpart

+partpart

+partpart

=rrrr

r由由rr r+drr+dr該質點的速度改變該質點的速度改變

先由一個質點切入再延伸到所有質點先由一個質點切入再延伸到所有質點

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

38

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 3535

tV

zVw

yVv

xVu

dtVda

wdt

dzvdt

dyudt

dxt

Vdt

dzz

Vdt

dyy

Vdt

dxx

VdtVda

AAA

AA

AA

AA

AA

AA

AA

AAAAAAAAA

partpart

+partpart

+partpart

+partpart

==rArr

===

partpart

+partpart

+partpart

+partpart

==

rrrrrr

rrrrrr

zVw

yVv

xVu

tVa

partpart

+partpart

+partpart

+partpart

=rrrr

r

Valid for any particlehellip

dividedividedtdt

套用到所有質點把下標套用到所有質點把下標AA拿掉拿掉

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

39

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 4545

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

partpart

+partpart

+partpart

+partpart

=

A shorthand notationA shorthand notationDt

VDar

r=

Scalar components 寫成分量型態寫成分量型態

簡寫成簡寫成helliphellip

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

40

Acceleration Field Acceleration Field －－Material DerivativeMaterial Derivative 5555

is termed the material derivative or is termed the material derivative or substantial derivativesubstantial derivative

DtVDar

r=

))(V(t)(

z)(

wy)(

vx)(

ut)(

Dt)(D

nablasdot+partpart

=

partpart

+partpart

+partpart

+partpart

=

r

Where the operatorWhere the operatorDt

)(D

實質導數實質導數

Material derivative is Material derivative is used to describe time used to describe time rates of change for rates of change for given particlegiven particle

物理意義物理意義

Time derivativeTime derivative

Spatial derivativeSpatial derivative

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

41

Physical SignificancePhysical Significance

zVw

yVv

xVu

tV

tDVDa

partpart

+partpart

+partpart

+partpart

==rrrrr

r

TotalTotalAccelerationAcceleration Convective AccelerationConvective Acceleration

LocalLocalAccelerationAcceleration

V)V(tV

tDVDa

rvrr

rnablasdot+

partpart

==

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

42

Physical SignificancePhysical Significance-- Unsteady EffectUnsteady Effect

Time derivativeTime derivative Local derivative Local derivativeIt represents effect of the unsteadiness of the flowIt represents effect of the unsteadiness of the flow

t)(

partpart

Local derivative acceleration

實質導數非穩定項實質導數非穩定項

實質導數非穩定項流體特性的非穩定效應實質導數非穩定項流體特性的非穩定效應沒有沒有涉及到位置改變屬於特性在原地的時間改變率涉及到位置改變屬於特性在原地的時間改變率

Flow fieldFlow field的其他特性或參數的其他特性或參數

Unsteady flowUnsteady flow

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

43

Physical SignificancePhysical Significance-- Convective EffectConvective Effect

Spatial derivativeSpatial derivative Convective derivative Convective derivativeIt represents the fact that a flow property associated with a It represents the fact that a flow property associated with a fluid particle may vary because of the motion of the fluid particle may vary because of the motion of the particle from one point in space to another pointparticle from one point in space to another point

Convective derivative acceleration))(V( nablasdotv

實質導數對流項流體特性由一位置移動到另一位置的實質導數對流項流體特性由一位置移動到另一位置的位置改變過程中所引發的效應位置改變過程中所引發的效應

實質導數對流項實質導數對流項

Flow fieldFlow field的其他特性或參數的其他特性或參數

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

44

For Various Fluid ParametersFor Various Fluid Parameters

TVtT

zTw

yTv

xTu

tT

DtDT

dtdz

zT

dtdy

yT

dtdx

xT

tT

dtTd AAAAAAAA

nablasdot+partpart

=partpart

+partpart

+partpart

+partpart

=rArr

partpart

+partpart

+partpart

+partpart

=

r

The material derivative concept is very useful in analysis involving various parameter not just the accelerationFor example consider a temperature field T=T(xyzt) associated with a given flow We can apply the chain rule to determine the rate of change of temperature as

個別質點個別質點

套到所有質點整個溫度場套到所有質點整個溫度場

用於用於Flow fieldFlow field的其他特性或參數的其他特性或參數

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

45

Example 44 Acceleration along a Example 44 Acceleration along a StreamlineStreamline

An incompressible inviscid fluid flows steadily past a sphere of radius R as shown in Figure E44a According to a more advancedanalysis of the flow the fluid velocity along streamline A-B is given by

where V0 is the upstream velocity far ahead of the sphere Determine the acceleration experienced by fluid particles as they flow along this streamline

i)xR(1V i u(x) V 3

3

0 +==

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

46

Example 44 Example 44 SolutionSolution

i xuu

tu

xuu

tV a ⎟

⎠⎞

⎜⎝⎛

partpart

+partpart

=partpart

+partpart

= 0 a 0 a xuu

tu a zyx ==

partpart

+partpart

=

( )[ ]4-303

3

0x 3X-RVXR1V

xuu a ⎟

⎠⎞

⎜⎝⎛ +=

partpart

=

( ) ( )( )4

32

0x RxxR1RV3a +

minus=

The acceleration along streamline The acceleration along streamline AA--BB

partpartu u partparttt=0 With the given velocity distribution along the streamline =0 With the given velocity distribution along the streamline the acceleration becomesthe acceleration becomes

oror

oror

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

47

Example 45 Acceleration from a Given Example 45 Acceleration from a Given Velocity FieldVelocity Field

Consider the steady twoConsider the steady two--dimensional flow field discussed in dimensional flow field discussed in Example 42Example 42 Determine the acceleration field for this flow Determine the acceleration field for this flow

)jyix)(V(V 0

rrl

rminus=

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

48

Example 45 Example 45 SolutionSolution1212

( )( )zVw

yVv

xVu

tVVV

tV

DtVDa

partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

==rrrr

rrrr

r

In general the acceleration is given byIn general the acceleration is given by

u = u = ((VV00 ll )x and v = )x and v = -- (V(V00 ll )y)yFor steady twoFor steady two--dimensional flowdimensional flow

jyvv

xvui

yuv

xuu

yVv

xVua

rrrr

r⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

+⎟⎟⎠

⎞⎜⎜⎝

⎛partpart

+partpart

=partpart

+partpart

=

jV)y(V)0)(x(Vi)0)(y(VV)x(Va 000000r

lll

r

lll

r⎟⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛minus⎟

⎠⎞

⎜⎝⎛minus+⎟

⎠⎞

⎜⎝⎛+⎟

⎠

⎞⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

2

20

y2

20

xyVaxVa

ll==

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

49

Example 45 Example 45 SolutionSolution1212

For this flow the magnitude of the acceleration is constant on cFor this flow the magnitude of the acceleration is constant on circles ircles centered at the origincentered at the origin

Also the acceleration vector is oriented at an angle Also the acceleration vector is oriented at an angle θθ from the x from the x axis whereaxis where

tan y

x

a ya x

θ = =

21222

0212z

2y

2x )yx(V)aaa(a +⎟

⎠⎞

⎜⎝⎛=++=l

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

50

Example 46 The Material DerivativeExample 46 The Material Derivative

A company produces a perishable product in a factory located at A company produces a perishable product in a factory located at x=0 and sells the product along the distribution route x gt 0 Thx=0 and sells the product along the distribution route x gt 0 The e selling price of the product P is a function of the length of tselling price of the product P is a function of the length of time after ime after it was produced t and the location at which it is sold x Thit was produced t and the location at which it is sold x That is at is P=P(xt) At a given location the price of the product decreasesP=P(xt) At a given location the price of the product decreases in in time (it is perishable) according to time (it is perishable) according to δδPPδδt= t= --8 dollarshr In 8 dollarshr In addition because of shipping costs the price increase with distaddition because of shipping costs the price increase with distance ance from the factory according to from the factory according to δδPPδδxx=02 dollarsmi If the =02 dollarsmi If the manufacturer wishes to sell the product for the same 100manufacturer wishes to sell the product for the same 100--dollar dollar price anywhere along the distribution route determine how fast price anywhere along the distribution route determine how fast he he must travel along the routemust travel along the route

實質導數的應用實質導數的應用

價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響價格受保存時間（產品亦腐敗的原因）與運輸距離（成本）的影響為了讓價格維持不變找出最佳的運輸速度為了讓價格維持不變找出最佳的運輸速度

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

51

Example 46 Example 46 SolutionSolution1212

xPu

tP

zPw

yPv

xPu

tPPV

tP

DtDP

partpart

+partpart

=partpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=r

iuV =r

For a given batch of the product (For a given batch of the product (LagrangianLagrangian description ) the time description ) the time rate of change of the price can be obtained by using the materiarate of change of the price can be obtained by using the material l derivativederivative

The motion is oneThe motion is one--dimensional with dimensional with

Where u is the speed at which the Where u is the speed at which the product is product is convectedconvected along its route along its route

Figure E46

讓價格維持不變讓價格維持不變

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

52

Example 46 Example 46 SolutionSolution2222

Thus the correct delivery speed isThus the correct delivery speed is

hrmi40midollars20

hrdollars8xPtPu ==

partpartpartpartminus

=

The price is to remain constant as the product moves along the The price is to remain constant as the product moves along the distribution route thendistribution route then

0xPu

tPor0

DtDp

=partpart

+partpart

=

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

53

Streamline Coordinates Streamline Coordinates 1515

Streamline coordinate is a coordinate system defined in Streamline coordinate is a coordinate system defined in terms of the streamline of the flowsterms of the streamline of the flowsUnit vectors are denoted byUnit vectors are denoted by

nandsrr

StreamlineStreamline的切線與法線單位向量的切線與法線單位向量

The flow plane is covered The flow plane is covered by an orthogonal curved by an orthogonal curved net of coordinate linesnet of coordinate lines

正交正交streamlinestreamline非直線非直線

兩者正交兩者正交

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

54

Streamline Coordinates Streamline Coordinates 2525

Why using the streamline coordinatesWhy using the streamline coordinatesThe velocity is always tangent to the s directionThe velocity is always tangent to the s direction

sVVrr

=This allows simplification in describing This allows simplification in describing the fluid particle acceleration and in the fluid particle acceleration and in solving the equations governing the solving the equations governing the flowflow

使用使用streamline coordinatestreamline coordinate的優點讓速度與加速的表達的優點讓速度與加速的表達簡單化速度只有切線方向加速度則有切線與法線方向簡單化速度只有切線方向加速度則有切線與法線方向

何以要使用何以要使用streamline coordinatesstreamline coordinates

nasaDt

VDa nsrr

rr

+==

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

55

Streamline Coordinates Streamline Coordinates 3535

If the streamlines are curved both of the speed of the particleIf the streamlines are curved both of the speed of the particle and its and its direction of flow are defineddirection of flow are defined

For a given particle the value of s changes with time but the For a given particle the value of s changes with time but the value value of n remains fixed because the particle flows along a streamlineof n remains fixed because the particle flows along a streamlinedefined by n=constantdefined by n=constantApplication of the chain rule gives Application of the chain rule gives

( )⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

+⎟⎠⎞

⎜⎝⎛

partpart

+partpart

+partpart

=+==dtdn

ns

dtds

ss

tsVs

dtdn

nV

dtds

sV

tV

DtsDVs

tDDV

DtsVDa

rrrr

rr

rr

)ns(ss)ns(VV rr==

⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=ssVVs

sVVa

rrr

ss是隨時間改變而變化是隨時間改變而變化nn則是常數則是常數

質點永遠是沿著質點永遠是沿著streamlinestreamline前進前進不會穿越不會穿越streamlinestreamline

SteadySteady

Rn

ss

ss lim

0s

rrr

=δδ

=partpart

rarrδ

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

56

Streamline Coordinates Streamline Coordinates 4545

nasaDt

VDa nsrr

rr

+==Rn

ss

ss lim

0s

rrr=

δδ

=partpart

rarrδ

nRVs

sVV

ssVVs

sVVa

2 rrr

rr+⎟

⎠⎞

⎜⎝⎛

partpart

=⎟⎠⎞

⎜⎝⎛

partpart

+⎟⎠⎞

⎜⎝⎛

partpart

=

Normal to the fluid motion

Convective acceleration along the streamlinealong the streamline在在Steady stateSteady state前提下前提下Local acceleration = 0Local acceleration = 0就只剩下沿著就只剩下沿著streamlinestreamline的的對流加速度對流加速度

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

57

Streamline Coordinates Streamline Coordinates 5555

sVVas partpart

= The convective acceleration along the The convective acceleration along the streamlinestreamline

RVa

2

n = The centrifugal acceleration normal to The centrifugal acceleration normal to the fluid motionthe fluid motion

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

58

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

習慣的方法習慣的方法

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

59

Basic Laws for a SystemBasic Laws for a System-- Conservation of MassConservation of Mass

Conservation of MassConservation of MassRequiring that the mass M of the system be constantRequiring that the mass M of the system be constant

Where the mass of the systemWhere the mass of the system

0dt

dM

system=⎟

⎠⎞

VddmM)system(V)system(Msystem ρ== intint

一個方程式說盡質量守恆的意涵一個方程式說盡質量守恆的意涵

MM又是什麽又是什麽

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

60

Basic Laws for a SystemBasic Laws for a System-- NewtonNewtonrsquorsquos Second Laws Second Law

Newtonrsquos Second LawStating that the sum of all external force acting on the system is equal to the time rate of change of linear momentum of the system

Where the linear momentum of the systemWhere the linear momentum of the systemsystem

dtPdF ⎟⎟⎠

⎞=

rr

VdVdmVP)system(V)system(Msystem ρ== intintrrr

PP又是什麽又是什麽

外力外力動量的時間變化率動量的時間變化率

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

61

Basic Laws for a SystemBasic Laws for a System-- The Angular Momentum PrincipleThe Angular Momentum Principle

The Angular Momentum PrincipleThe Angular Momentum PrincipleStating that the rate of change of angular momentum is equal to the sum of all torques acting on the system

Where the angular momentum of the systemWhere the angular momentum of the system

Torque can be produced by surface and body forces and Torque can be produced by surface and body forces and also by shafts that cross the system boundaryalso by shafts that cross the system boundary

systemdtHdT ⎟⎟⎠

⎞=

rr

VdVrdmVrH)system(V)system(Msystem ρtimes=times= intint

rrvrr

shaft)system(Ms TdmgrFrTrrrrrr

+times+times= int

力矩力矩 角動量的時間變化率角動量的時間變化率

產生力產生力矩的來源矩的來源

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

62

Basic Laws for a SystemBasic Laws for a System-- The First Law of ThermodynamicsThe First Law of Thermodynamics

The First Law of Thermodynamics (Conservation of The First Law of Thermodynamics (Conservation of Energy)Energy)Requiring that the energy of system be constantRequiring that the energy of system be constant

dEWQ =δminusδ

Where the total energy of the systemWhere the total energy of the system

u is specific internal energy V the speed and z the u is specific internal energy V the speed and z the height of a particle having mass dmheight of a particle having mass dm

systemdtdEWQ ⎟

⎠⎞=minus ampamp

gz2

Vue

VdeedmE2

)system(V)system(Msystem

++=

ρ== intint

系統內能的改變系統內能的改變

系統吸收的熱量扣掉系統對外作的功系統吸收的熱量扣掉系統對外作的功

系統系統內能內能的增加與系統對外作的增加與系統對外作功功的總和等於系統吸收的熱量的總和等於系統吸收的熱量

能量守恆定律能量守恆定律

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

63

Basic Laws for a SystemBasic Laws for a System-- The Second Law of ThermodynamicsThe Second Law of Thermodynamics

The Second Law of ThermodynamicsThe Second Law of ThermodynamicsIf an amount of heat If an amount of heat δδQQ is transferred to a system at is transferred to a system at temperature T the change in entropy temperature T the change in entropy dSdS of the system of the system satisfiessatisfies

TQdS δ

ge

Where the total entropy of the systemWhere the total entropy of the system

QT1

dtdS

system

ampge⎟⎠⎞

VdssdmS)system(V)system(Msystem ρ== intint

系統熵的改變系統熵的改變

於溫度於溫度TT下傳下傳到系統的熱量到系統的熱量

方向定律方向定律

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

64

Second Law of Thermodynamics Second Law of Thermodynamics 1414

ClausiusClausius（克勞修斯（克勞修斯18221822--18881888））It is It is impossible that at the end of a cycle of changes impossible that at the end of a cycle of changes heat has been transferred from a colder to a hotter heat has been transferred from a colder to a hotter body without at the same time converting a certain body without at the same time converting a certain amount of work into heatamount of work into heat不可能把熱從低溫物體傳到高溫物體而不引不可能把熱從低溫物體傳到高溫物體而不引起其他變化起其他變化

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

65

Second Law of Thermodynamics Second Law of Thermodynamics 2424

Lord KelvinLord Kelvin（開爾文（開爾文18241824--19071907））In a cycle In a cycle of processes it is impossible to transfer heat from a of processes it is impossible to transfer heat from a heat reservoir and convert it all into work without heat reservoir and convert it all into work without at the same time transferring a certain amount of at the same time transferring a certain amount of heat from a hotter to a colder bodyheat from a hotter to a colder body不可能從單一熱源取出熱使之完全變為功而不可能從單一熱源取出熱使之完全變為功而不發生其他的變化不發生其他的變化

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

66

Second Law of Thermodynamics Second Law of Thermodynamics 3434

Ludwig Ludwig BoltzmannBoltzmann（波爾茲曼（波爾茲曼18441844--1906 1906 ））For an adiabatically enclosed system the entropy For an adiabatically enclosed system the entropy can never decrease Elements in a closed system can never decrease Elements in a closed system tend to seek their most probable distribution tend to seek their most probable distribution therefore a high level of organization is very therefore a high level of organization is very improbable improbable 絕熱系統中的熵值決不會降低絕熱系統中的熵值決不會降低

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

67

Second Law of Thermodynamics Second Law of Thermodynamics 4444

狀態函數狀態函數 Entropy S Entropy S

任意過程中任意過程中 dSdS 的改變都遵從下列關係的改變都遵從下列關係

『『 gtgt』』應用在自發的不可逆過程（應用在自發的不可逆過程（Spontaneous or Spontaneous or Irreversible processIrreversible process））

『『 ==』』只限於可逆的過程只限於可逆的過程（（Reversible processReversible process））

TQdS revδ

=

TQdS δ

ge

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

68

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

69

Method of Analysis

System methodSystem methodIn mechanics coursesIn mechanics coursesDealing with an easily identifiable rigid bodyDealing with an easily identifiable rigid body

Control volume methodControl volume methodIn fluid mechanics courseIn fluid mechanics courseDifficult to focus attention on a fixed identifiable Difficult to focus attention on a fixed identifiable quantity of massquantity of massDealing with the flow of fluidsDealing with the flow of fluids

討論的標的是一個容易界定的剛體容討論的標的是一個容易界定的剛體容易從環境區隔出來的剛體如材料力易從環境區隔出來的剛體如材料力學工程力學等學科學工程力學等學科

針對難以掌握針對難以掌握很難把它從環境很難把它從環境區隔出來的標的如區隔出來的標的如流體力學流體力學

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

70

System MethodSystem Method

A system is defined as a fixed identifiable quantity of massThe boundaries separate the system from the surroundingThe boundaries of the system may be fixed or movable No mass crosses the system boundaries

定義定義「「SystemSystem」固定且易於界定的量體」固定且易於界定的量體有一明顯邊界與周邊環境切割有一明顯邊界與周邊環境切割

「「SystemSystem」與周邊環境的邊界可以是固定的」與周邊環境的邊界可以是固定的或移動的惟沒有或移動的惟沒有massmass可以貫穿邊界可以貫穿邊界

See Page 70See Page 70

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

71

System Representation System Representation 1212

A system is a collection of matter of fixed identity which A system is a collection of matter of fixed identity which may move flow and interact with its surroundingsmay move flow and interact with its surroundingsA system is a specific identifiable quantity of matter It A system is a specific identifiable quantity of matter It may consist of a relatively large amount of mass or it may may consist of a relatively large amount of mass or it may be an infinitesimal sizebe an infinitesimal sizeIn the study of statics and dynamics the In the study of statics and dynamics the freefree--body body diagram conceptdiagram concept is used to identify an object and isolate it is used to identify an object and isolate it from its surroundings replace its surroundings by the from its surroundings replace its surroundings by the equivalent actions that they put on the object equivalent actions that they put on the object

在固體力學領域中在固體力學領域中我們容易理解如何界定我們容易理解如何界定「「SystemSystem」然對象是流體時」然對象是流體時helliphelliphelliphellip

在應用力學與材料力學中相當熟悉的語言在應用力學與材料力學中相當熟悉的語言

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

72

System Representation System Representation 2222

A mass of air drawn into an air compressor can be A mass of air drawn into an air compressor can be considered as a system It changes shape and size its considered as a system It changes shape and size its temperature may changetemperature may changeIt is difficult to identify and keep track of a specific It is difficult to identify and keep track of a specific quantity of matter associated with the original air quantity of matter associated with the original air drawn into the compressordrawn into the compressor當一團空氣被抽進去壓縮機時當一團空氣被抽進去壓縮機時若把該團空氣被視為若把該團空氣被視為「「SystemSystem」」則該團空氣不僅在過程中發生形狀改則該團空氣不僅在過程中發生形狀改變也發生尺度大小的改變甚至溫度也改變變也發生尺度大小的改變甚至溫度也改變如何去界定與追蹤一定量體的空氣如何去界定與追蹤一定量體的空氣

Control Volume methodControl Volume method

當對象是空氣時麻煩可就來了當對象是空氣時麻煩可就來了helliphellip

只好求助於另一種方法了只好求助於另一種方法了helliphellip

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

73

Control Volume MethodControl Volume Method

Control volume is an arbitrary volume in space through which the fluid flowsThe geometric boundary of the control volume (CV) is called the ldquoControl Surface (CS)rdquoThe CS may be real or imaginaryThe CV may be at rest or in motion

Control volumeControl volume的邊界叫的邊界叫control control surfacesurface可能是實體也可能是可能是實體也可能是虛幻的虛幻的CVCV可以是固定的可以可以是固定的可以是移動的更可以是變形的是移動的更可以是變形的

CVCV是空間中任一量體虛的實的流體可以進出是空間中任一量體虛的實的流體可以進出

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

74

Control Volume Representation Control Volume Representation 1414

A control volume is a volume in space through which fluid A control volume is a volume in space through which fluid may flowmay flowExample Determining the Example Determining the forces put on a fan airplane or forces put on a fan airplane or automobileautomobile by control volume approachby control volume approachIdentify a specific volume in space (a volume Identify a specific volume in space (a volume associated with the fan airplane or automobile) and associated with the fan airplane or automobile) and analyze the fluid flow within through or around that analyze the fluid flow within through or around that volumevolumeThe control volume can be a fixed moving or The control volume can be a fixed moving or deforming volumedeforming volume

Control volumeControl volume是空間中因為流體分析所需設定是空間中因為流體分析所需設定出來的出來的volumevolume被分析的標的可以經由被分析的標的可以經由CSCS進出進出CVCV

將將fanfanairplaneairplane或或automobileautomobile視為空間視為空間中中特定特定的的volumevolume

分析分析volumevolume內部四周流經內部四周流經volumevolume的流體的流體

利用利用CVCV方法分析方法分析fanfanairplaneairplane等的推力等的推力helliphellip

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

75

Control Volume Representation Control Volume Representation 2424

For case (a) fluid flows through a pipe The fixed control For case (a) fluid flows through a pipe The fixed control surface consists of the inside surface of the pipe the outlet surface consists of the inside surface of the pipe the outlet end at section (2) and a section across the pipe at (1)end at section (2) and a section across the pipe at (1)

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

76

Control Volume Representation Control Volume Representation 3434

For case (b) control volume is the rectangular volume For case (b) control volume is the rectangular volume surrounding the jet engine If the airplane to which the surrounding the jet engine If the airplane to which the engine is attached is sitting still on the runway air flow engine is attached is sitting still on the runway air flow through this control volume At time t=tthrough this control volume At time t=t11 the air was the air was within the engine itself At time t=twithin the engine itself At time t=t22 the air has passed the air has passed through the engine and is outside of the control volume through the engine and is outside of the control volume At this latter time other air is within the engineAt this latter time other air is within the engine

飛飛機機仍仍留留在在跑跑道道時時

Control volumeControl volume涵蓋引擎涵蓋引擎 tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

77

Control Volume Representation Control Volume Representation 4444

For case (c) the deflating balloon provides an example of For case (c) the deflating balloon provides an example of a deforming control volume The control volume (whose a deforming control volume The control volume (whose surface is the inner surface of the balloon) decreases in surface is the inner surface of the balloon) decreases in size size

CSCS包括包括pipepipe的內徑的內徑

fluid flows through a pipefluid flows through a pipe

Control volumeControl volume涵蓋引擎涵蓋引擎

tt22時剛才在時剛才在CVCV的空氣跑到的空氣跑到CVCV外外頭此時在頭此時在CVCV內的是另外一團空氣內的是另外一團空氣

tt11時空氣在時空氣在CVCV內內

形狀可變的ＣＶ形狀可變的ＣＶ

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

78

System and Control Volume System and Control Volume RepresentationRepresentation

BASIC LAWS

System Method Control Volume MethodControl Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Governing EquationControl Volume Formulation

Conservation of MassConservation of MassNewtonNewtonrsquorsquos Second Laws Second LawThe Angular Momentum PrincipleThe Angular Momentum PrincipleThe First Law of ThermodynamicsThe First Law of ThermodynamicsThe Second Law of ThermodynamicsThe Second Law of Thermodynamics

開闢捷徑開闢捷徑Reynolds Transport TheoremReynolds Transport Theorem

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

79

Analytic ToolAnalytic Tool

Reynolds Transport TheoremReynolds Transport Theorem

Shift from system representation to Shift from system representation to control volume representationcontrol volume representation

The Reynolds transport theorem provides the relationship The Reynolds transport theorem provides the relationship between the time rate of change of an extensive property between the time rate of change of an extensive property for a system and that for a control volumefor a system and that for a control volume

建立建立system method system method 與與control volume methodcontrol volume method下「下「 time rate time rate of change of an extensive property of change of an extensive property 」的關係」的關係

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

80

Reynolds Transport Theorem Reynolds Transport Theorem 1414

All physical laws are stated in termed of various physical parameters Velocity acceleration mass temperature and momentum are but a few of the more parametersLet BB represent any one of the system extensive extensive propertiesproperties ( mass linear momentum angular momentum energy and entropy) the corresponding intensive intensive property property (extensive property per unit mass ) will be designated by bb）

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

所有所有physical lawsphysical laws提到的提到的physical parametersphysical parameters可以分成可以分成helliphellip

intensive propertyintensive property 質量質量 extensive propertiesextensive properties

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

81

Reynolds Transport Theorem Reynolds Transport Theorem 2424

If B=m (Mass) b=1If B=mV22 (Kinematic energy of the mass) b= V22 If B=mV (Momentum of the mass) b=V

( ) intsum ρequivδρ=rarrδ

VbdVbB sysiii0V

SYS limmbB =

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

82

Reynolds Transport Theorem Reynolds Transport Theorem 3434

Most of the laws governing fluid motion involve the time Most of the laws governing fluid motion involve the time rate of change of an extensive property B of a fluid rate of change of an extensive property B of a fluid system system ndashndash the rate at which the momentum of a system the rate at which the momentum of a system changes with time the rate at which the mass of a system changes with time the rate at which the mass of a system changes with time and so onchanges with time and so on

( )dt

Vdbd

dtdB syssys ρ

= int

Time rate of change of an extensive Time rate of change of an extensive property B of a fluid systemproperty B of a fluid system

Ｂ可能是Ｂ可能是massmassmomentummomentum

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

83

Reynolds Transport Theorem Reynolds Transport Theorem 4444

To formulate the law into a control volume we must To formulate the law into a control volume we must obtain an expression for the time rate of change of an obtain an expression for the time rate of change of an extensive property within a control volume extensive property within a control volume BBcvcv not not within a systemwithin a system

( )dt

Vbdd

dtdB cvcv int ρ

=

dt

dBdt

dB cvsys rarr

The Reynolds transport theorem The Reynolds transport theorem provides the relationship between the provides the relationship between the time rate of change of an extensive time rate of change of an extensive property for a system and that for a property for a system and that for a control volumecontrol volume

Time rate of change of an extensive Time rate of change of an extensive property B with a control volumeproperty B with a control volume

工作目標工作目標

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

84

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem1919

How to derive a control volume representation from a How to derive a control volume representation from a system representation of a fluid flowsystem representation of a fluid flow

Choosing a fixed control volume in space relative to Choosing a fixed control volume in space relative to coordinate xyzcoordinate xyzImaging selecting an arbitrary piece of the flowing Imaging selecting an arbitrary piece of the flowing fluid at time = t and dyeing this piece of fluidfluid at time = t and dyeing this piece of fluidThe initial shape of the flowing fluid is chosen as The initial shape of the flowing fluid is chosen as control volumecontrol volumeAfter an infinitesimal time After an infinitesimal time δδtt the flowing fluid will the flowing fluid will have moved to a new locationhave moved to a new location

system representation system representation CV representationCV representation

步步驟驟

在時間在時間ttflowing fluid flowing fluid 初始形狀與初始形狀與CVCV一致一致

一小段一小段時間時間後後flowing fluidflowing fluid移動另一新位置移動另一新位置

相對直角座標系選一固定相對直角座標系選一固定CVCV

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

85

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem2929

At t At t BBSYSSYS(t(t) ) ≣≣BBCVCV(t(t))At At t+t+δδtt BBSYSSYS ((t+t+δδtt ) ) ≣≣BBCVCV ((t+t+δδtt ) ) -- BBⅠⅠ ((t+t+δδtt ) + B) + BⅡⅡ ((t+t+δδtt ))

( ) ( ) ( ) ( ) ( )t

ttBt

ttBt

tB)tt(Bt

tBttBt

B cvcvsyssyssys

δδ+

+δ

δ+minus

δminusδ+

=δ

minusδ+=

δ

δΙΙΙ

The change in the amount of B in the system in the time intervalThe change in the amount of B in the system in the time interval δδtt

1 2 3

想像有一團待追蹤的想像有一團待追蹤的flowing fluidflowing fluid（（systemsystem））

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

86

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem3939

In the limit In the limit δδttrarrrarr00 evaluate each of the three terms evaluate each of the three termshelliphellip1 The first term is seen to be the time rate of change of the The first term is seen to be the time rate of change of the

amount of B within the control volumeamount of B within the control volume

int ρpartpart

=part

part=

δminusδ+

rarrδ

CVCV

CVCV

0t

Vdbtt

Bt

)t(B)tt(BlimCVCV內內BB對時間改變率對時間改變率

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

87

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem4949

1111I

0tin

1111111I

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

2222II

0tout

2222222II

VAbt

)tt(BlimB

tVAb)V)(b()tt(B

ρ=δ

δ+=

δρ=δρ=δ+

rarrδamp

BB流入流入CCVV的的flow rateflow rate

BB流出流出CVCV的的flow rateflow rate

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

88

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem5959

The relationship between the time rate of change of B for The relationship between the time rate of change of B for the system and that for the control volumethe system and that for the control volume

This is a version of the Reynolds transport theorem This is a version of the Reynolds transport theorem valid under the restrictive assumptions associated with valid under the restrictive assumptions associated with the flow through a pipethe flow through a pipe

11112222CVsys

inoutCVsys

bVAbVAt

Bdt

dBor

B-Bt

Bdt

dB

ρminusρ+part

part=

+part

part= ampamp

For more general conditionFor more general condition

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

89

Example 47 Time Rate of Change for a Example 47 Time Rate of Change for a System and a Control VolumeSystem and a Control Volume

Find flows from the fire extinguisher tank shown in Figure E47Find flows from the fire extinguisher tank shown in Figure E47Discuss the differences between Discuss the differences between dBdBsyssysdtdt and and dBdBcvcvdtdt if B represents if B represents massmass

Figure E47

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

90

Example 47 Example 47 SolutionSolution1212

dt

dVd

dtdm

dtdB syssyssys

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

With B=m the system mass it follows that b=1With B=m the system mass it follows that b=1

AndAnd

dt

dVd

dtdm

dtdB cvcvcv

⎟⎠⎞⎜

⎝⎛ intρ

=equiv

dt

Vdbd

dtdB syssys

⎟⎠⎞⎜

⎝⎛ ρ

=int

dt

Vbdd

dtdB cv

cv⎟⎠⎞⎜

⎝⎛ ρ

=int

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

91

Example 47 Example 47 SolutionSolution2222

If mass is to be conserved (one of the basic laws governing fluiIf mass is to be conserved (one of the basic laws governing fluid d motion) the mass of the fluid in the system is constant so thmotion) the mass of the fluid in the system is constant so that at

0dt

dVdsys =

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint

On the other hand it is equally clear that some of the fluid haOn the other hand it is equally clear that some of the fluid has left the s left the control volume through the nozzle on the tank Hence the amountcontrol volume through the nozzle on the tank Hence the amount of of mass within the tank (the control volume) decreases with time omass within the tank (the control volume) decreases with time orr

0dt

dVdcv lt

⎟⎟⎠

⎞⎜⎜⎝

⎛ρint Clearly the meanings of Clearly the meanings of dBdBsyssysdtdt and and

dBdBcvcvdtdt are different For this example are different For this example dBdBsyssysdtdt lt lt dBdBcvcvdtdt Other situations Other situations may have may have dBdBsyssysdtdt ≧≧dBdBcvcvdtdt

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

92

Example 48 Use of the Reynolds Example 48 Use of the Reynolds Transport TheoremTransport Theorem

Consider again the flow from the fire extinguisher shown in FiguConsider again the flow from the fire extinguisher shown in Figure re E47 Let the extensive property of interest be the system mass E47 Let the extensive property of interest be the system mass ((BB = = mm the system mass or the system mass or bb =1)and write the appropriate form of =1)and write the appropriate form of the Reynolds transport theorem for the flow the Reynolds transport theorem for the flow

Figure E47

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

93

Example 48 Example 48 SolutionSolution1212

For this case there is no inlet section (1) across which the fFor this case there is no inlet section (1) across which the fluid flows luid flows into the control volume (into the control volume (AA1 1 =0) =0) There is however an outlet section There is however an outlet section (2) Thus the Reynolds transport theorem can be written as(2) Thus the Reynolds transport theorem can be written as

The basic law of The basic law of conservation of massconservation of mass

( )222

cv VAt

dVρminus=

part

ρpart int

( )222

cvsys VAt

dV

dtmd

ρ+part

ρpart= int

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

94

Example 48 Example 48 SolutionSolution2222

When there were both an inlet and an outlet to the control volumWhen there were both an inlet and an outlet to the control volumee

For steady flowFor steady flow

( )222111

cv VAVAt

dVρminusρ=

part

ρpart int

222111 VAVA ρ=ρ

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

95

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem6969

For For much more generalmuch more general conditionsconditions

A general fixed control volume with fluid flow through it

Control volume and system for flow through an arbitrary fixed control volume

CVCV Flowing fluidFlowing fluid

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

96

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem7979

The second term is inflow rate of B into the control volumeThe second term is inflow rate of B into the control volume2

intintint sdotρminus=θρminus==ininin CSCSCS inin dAnVbdAcosbVBdB rr

ampamp

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

97

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem8989

intintint sdotρ=θρ==

θδρ=δ

δθδρ=

δδρ

=δrarrδrarrδ

outoutout CSCSCS outout

0t0tout

dAnVbdAcosbVBdB

AcosbVt

A)tcosbV(limtVblimB

rrampamp

amp

3 The third term is outflow rate of B from the control volumeThe third term is outflow rate of B from the control volume

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

98

Possible Velocity ConfigurationPossible Velocity Configuration

Possible velocity configurations on portions of the control surface (a) inflow (b) no flow across the surface (c) outflow

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

99

Derivation of Reynolds Transport TheoremDerivation of Reynolds Transport Theorem9999

The net flux (The net flux (flowrateflowrate) of parameter B across the entire ) of parameter B across the entire control surface iscontrol surface is

int sdotρ=minus dAnVbBB CSinoutrr

ampamp

dAnVbVbdt

dAnVbt

Bdt

dB

CSCV

CSCVsys

rr

rr

sdotρ+ρpartpart

=

sdotρ+part

part=

intint

int

The Reynolds Transport TheoremThe Reynolds Transport TheoremThis is the fundamental relation between the rate of change of aThis is the fundamental relation between the rate of change of any ny arbitrary extensive property B of a system and the variations arbitrary extensive property B of a system and the variations of of this property associated with a control volumethis property associated with a control volume

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

100

Physical InterpretationPhysical Interpretation

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B of the system This may represent the extensive property B of the system This may represent the rate of change of rate of change of mass momentum energy or angular mass momentum energy or angular momentummomentum of the systemof the system

is the rate of change of any arbitrary is the rate of change of any arbitrary extensive property B extensive property B withinwithin the control volume at a given the control volume at a given timetime

is the net rate of flux of extensive pro is the net rate of flux of extensive property perty B out through the control surfaceB out through the control surface

int ρpartpart

CVVdb

t

int sdotρCS

AdVbrr

system)dtdB

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

101

Relationship with Material DerivativeRelationship with Material Derivative

( ) ( ) ( ) ( ) ( ) ( ) ( )z

wy

vx

ut

VtDt

Dpartpart

+partpart

+partpart

+partpart

=nablasdot+partpart

=

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Convective effectThe effect associated with the particlersquos motion

Unsteady effect

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

102

Moving Control VolumeMoving Control Volume1212

Example of a moving control volumeExample of a moving control volume

VVcvcv is the velocity of control volume V is the absolute is the velocity of control volume V is the absolute velocity measured relative to inertial coordinate systemvelocity measured relative to inertial coordinate systemW is the relative velocity measured relative to the W is the relative velocity measured relative to the moving control volume moving control volume ndashndash the fluid velocity seen by an the fluid velocity seen by an observer riding along on the control volumeobserver riding along on the control volume

CV

CV

CV

VVW

WVV

WVV

rrr

rrr

rrr

minus=

+=

minus=

站在固定座標看到的流體速度（絕對）

站在固定座標看到的CV移動速度 觀察者站在CV上看

到的流體速度

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

103

Moving Control VolumeMoving Control Volume2222

The Reynolds transport equation for a control volume The Reynolds transport equation for a control volume moving with moving with constant velocityconstant velocity isis

dAnWbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

Control volume and system as Control volume and system as seen by an observer moving seen by an observer moving with the control volumewith the control volume

dAnVbVbdtdt

dBCSCV

sys rrsdotρ+ρ

partpart

= intint

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection

104

Selection of a Control VolumeSelection of a Control Volume

To ensure that the points associated with unknown To ensure that the points associated with unknown parameters are located on the control surface not buried parameters are located on the control surface not buried within the control volumewithin the control volumeIf possible the control surface should be normal to the If possible the control surface should be normal to the fluid velocityfluid velocityNot wrong but much better selection Not wrong but much better selection