OUTLINE FOR Chapter 2

REVIEW OF VECTOR RELATIONS (I)

AERODYNAMICS (W1-2-2)

REVIEW OF VECTOR RELATIONS (II)• Scalar Fields (pressure, density, temperature..)

• Vector Fields (velocity..)

• Scalar and Vector Products

• Gradients of Scalar Fields

1. Its magnitude is the maximum rate of change of p per unitlength of the coordinate space at the given point.2. Its direction is that of the maximum rate of change of p atthe given point.

• Directional derivatives:


Cartesian:

Cylinerical: Spherical:

REVIEW OF VECTOR RELATIONS (III)• Divergence of a Vector Field => Scalar

The physical meaning of divergence is the rate of change of the volume of a moving fluid element, per unit volume. <== see Chapter 2.3 for detail!

• Curl of a Vector Field => Vector

The physical meaning of curl is twice of the angular velocity vector of a fluid element.

AERODYNAMICS (W1_2_5)

REVIEW OF VECTOR RELATIONS (IV)

• Line integral

• Surface integral

• Volume integral

Relations between Line, Surface and Volume integrals:

Stokes theorem

Consider a vector field

Divergence theorem

Gradient theorem

• Line integral to Surface integral

• Surface integral to Volume integral

METHODS OF ANALYSIS

• System method In mechanics courses.Dealing with an easily identifiable rigid body.

• Control volume method In fluid mechanics course.Difficult to focus attention on a fixed identifiable

quantity of mass.Dealing with the flow of fluids.

System Method• A system is defined as a fixed, identifiable quantity of

mass.• The boundaries separate the system from the surrounding.• The boundaries of the system may be fixed or movable. No

mass crosses the system boundaries.

Piston-cylinder assembly: The gas in the cylinder is the system. If the gas is heated, the piston will lift the weight;The boundary of the system thus move.Heat and work may cross the boundaries, but the quantity of matter remain fixed.

Control Volume Method• Control Volume (CV) is an arbitrary volume in space

through which the fluid flows.• The geometric boundary of the control volume is called the

Control Surface (CS).

• The CS may be real or imaginary.• The CV may be at rest or in motion.

METHODS OF DESCRIPTION

• Lagrangian description => System• Eulerian description => Control volume

Lagrangian Description

• The fluid particle is colored, tagged or identified.• Determining how the fluid properties associated with the

particle change as a function of time. Example: one attaches the temperature-measuring device

to a particular fluid particle A and record that particle’s temperature as it moves about. TA = TA(t)=T (xo,yo,zo, t)where particle A passed through coordinate (xo,yo,zo) at to

The use of may such measuring devices moving with various fluid particles would provide the temperature of these fluid particles as a function of time.

• Attention is focused on a material volume (MV) and follow individual fluid particle as it move.

Eulerian Description• Attention is focused on the fluid passing through a

control volume (CV) fixed in the space.• Obtaining information about the flow in terms of what

happens at the fixed points in space as the fluid flows past those points.

• The fluid motion is given by completely prescribing the necessary properties as a functions of space and time.

Example: one attaches the temperature-measuring device to a particular point (x,y,z) and record the temperature at that point as a function of time. T = T ( x , y , z , t ) => field concept.The independent variables are the spatial coordinates ( x , y , z) and time t

Field Representation of flow• At a given instant in time, any fluid property ( such as density,

pressure, velocity, and acceleration) can be described as a functions of the fluid’s location.This representation of fluid parameters as functions of the spatial coordinates is termed a field representation of flow.

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

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

ktzyxwjtzyxvitzyxuV ),,,( ),,,( ),,,( ++=

Ball, with mass m, released from rest at a height y= 0. Air resistance is neglected FD=0. (a) Find the Lagrangian description of this velocity field. (b) Find the Eulerian description of this velocity field.

y

Basic Laws

• Conservation of mass – Continuity Equation• Conservation of (angular) Momentum

- Newton’s second law of motion.• Conservation of Energy

The first law of thermodynamics

Analysis of any problem in fluid mechanics necessarily includes statement of the basic laws governing the fluid motion. The basic laws, which applicable to any fluid, are:

BASIC LAWS FOR A SYSTEM- Conservation of Mass

• Conservation of Mass Requiring that the mass, M, of the system be constant.

Where the mass of the system

0)(

=== ∫ systemV

system

dVDtD

DtDM

dtdM ρ

BASIC LAWS FOR A SYSTEM- Conservation of Momentum

• Newton’s Second Law Stating 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 P is the linear momentum of the system

∫==

=

)(systemVsystem

dVVDtD

DtPD

dtPdF ρ

BASIC LAWS FOR A SYSTEM- Conservation of Energy

• The First Law of Thermodynamics Requiring that the energy of system be constant.

dEWQ =−δδ

Where E is the total energy of the system and et is the total energy of the system per unit mass

e is specific internal energy, V the speed, and z the height of a particle having mass dm.

)()(∫==

=−

systemV tsystem

dVeDtD

DtDE

dtdEWQ ρ

gzVeet ++=2

2

How To Derive Control Volume Formulation

BASIC LAWS

System Method Control Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Reynolds Transport TheoremIntegral (large) control volume

Differential (differential) control volume) Total (material) derivative

Transformation between Lagrangianand Eulerian Description

• It is more nature to apply conservation laws by using Lagrangian description (ie. Material Volume).

• However, the Eulerian description (ie. Control Volume) is preferred for solving most of problem in fluid mechanics.

• The two descriptions are related and there are a transformation formula called Reynolds transport theorem and material derivative between Lagrangian and Eulerian descriptions.

Reynolds Transport Theorem

∫∫∫∫∫∫∫∫ •+∂∂

=CSVCsystem

SdVVdt

VdDtD

ααα

By converting the surface integral to volume integral by use of Gauss theorem

∫∫∫∫∫ •∇=•VCCS

VdVSdV )(

αα

Langragian derivative of a volume integral of a given property

This is the fundamental relation between the rate of change of any arbitrary extensive property, α, of a system and the variations of this property associated with a control volume.

∫∫∫∫∫∫∫∫∫∫∫∫ •∇+∂∂

=•∇+∂∂

=VCVCVCsystem

VdVt

VdVVdt

VdDtD ))(()(

ααααα

Conservation of Mass• Basic Law for Conservation of Mass• The system and integral (large) control volume

formulation ---- Reynolds Transport Theorem

α=ρ

ααα∇•+

∂∂

= utDt

D

Material derivativeContinuity equation

A partial differential equation => velocity is continuous

0)( =•∇+∂∂ V

t

ρρ

0)()()( =•∇+=•∇+∇•+∂∂

=•∇+∂∂ V

DtDVV

tV

t

ρρρρρρρ

0=∫∫∫system

VdDtD ρ

∫∫∫∫∫∫ •∇+∂∂

=VCsystem

VdVt

VdDtD ))((

ααα

0))(( =•∇+∂∂

= ∫∫∫∫∫∫VCsystem

VdVt

VdDtD

ρρρ

How To Derive Control Volume Formulation

BASIC LAWS

System Method Control Volume Method

Governing EquationSystem Formulation

Governing EquationControl Volume Formulation

Reynolds Transport TheoremIntegral (large) control volume

Differential (differential) control volume) Total (material) derivative

Material Derivative (I)• Let α(x,y,z,t) be any field variable, e.g., ρ, T, V=(u,v,w), etc.

(Eulerian description)• Observe a fluid particle for a time period δt as it flows

(Langrangian description)• During the time period, the position of the fluid particle will

change by amounts δx , δy , δz, while its vale of α will change by an amount

• As one follow the fluid particle,So

which is called the material, total, or substantial derivative.

zz

yy

xx

tt

δαδαδαδαδα∂∂

+∂∂

+∂∂

+∂∂

=

),,(),,( wvutz

ty

tx

=δδ

δδ

δδ

zw

yv

xu

ttz

zty

ytx

xttDtD

t ∂∂

+∂∂

+∂∂

+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

==→

ααααδδα

δδα

δδαα

δδαα

δ 0lim

Material Derivative (II)• Use the notation D/Dt to emphasize that the

material derivative is the rate of change seen by an observer “following the fluid.”

• The material derivative express a Langrangianderivative in terms of Eulerian derivatives.

• In vector form,

ααααααα )( ∇•+∂∂

=∂∂

+∂∂

+∂∂

+∂∂

= Vtz

wy

vx

utDt

D

EXAMPLE OF SUBSTANTIAL DERIVATIVE

The velocity flow field of a steady state flow is given by the equations: u=-x ; v=y

The temperature of the field is described by the following expression: T(x,y,t)=xt+3xy

Determine the time rate of change of temperature of a fluid element as it passes through the point (1, -2) at time t=6.

the time rate of change of temperature of a fluid element

local derivative

convective derivative

yTv

xTu

tT

DtDT

∂∂

+∂∂

+∂∂

=

5)3)(2()66)(1(1)3()3)(( −=−+−−+=++−+=∂∂

+∂∂

+∂∂

= xyytxxyTv

xTu

tT

DtDT

x

y

Ball, with mass m, released from rest at a height y= 0. Air resistance is neglected FD=0. (a) From the Lagrangian description of this velocity field, find

the acceleration of the ball.(b) From the Eulerian description of this velocity field, , find the acceleration of the ball.

y

0=∇•+∂∂

= Mut

MDt

DM

Conservation of Mass• Basic Law for Conservation of Mass• The system and differential control volume

formulation ---- Material derivative

0=Dt

DM

α=M

0)()(=∇•+∇•+

∂∂

+∂∂

=∇•+∂

∂ ρρρρρρ uVVutVV

tVu

tV

0)()( =∇•+∂∂

+∇•+∂∂

=∇•+∇•+∂∂

+∂∂ Vu

tVu

tVuVVu

tVV

t ρρρρρρρ

0)()1( =•∇+=+=+ VDtD

DtVD

VDtD

DtVD

VDtD

ρρρρρρ

VM ρ=

Volume dilation = divergence of velocity field

ααα∇•+

∂∂

= utDt

D

Physical Meaning of V

•∇

CONSERVATION OF MASSRectangular Coordinate System

• The differential equation for conservation of mass:The continuity equation

By “Del” operator

The continuity equation becomes

0)()( =•∇+∂∂

=•∇+∇•+∂∂

=•∇+ Vt

VVt

VDtD

ρρρρρρρ

Description and Classification of Fluid Motions

Continuity Equation for Incompressible FluidDefinition of Incompressible fluid:As a given fluid is followed, not only will its mass be observed to remain constant, but its volume, and hence its density, will be observed to remain constant.

0)( =ρDtD

Continuity eq.

Continuity equation

Material derivative

Follow a fluid particle

01==•∇

DtVD

VV

?cosntant =ρ

0)( =•∇+∂∂

=•∇+ Vt

VDtD

ρρρρ

Stratified-Fluid Flow• A fluid particle along the line ρ1 or ρ2 will have its

density remain fixed at ρ=ρ1 or ρ=ρ2

0)( =ρDtD

Follow a fluid particleStratified-fluid flow is considered to be incompressible, but ρ is not constant (ρ≠constant ) everywhere ie. ə ρ/ ə x ≠0, ə ρ/ əy≠0,

Stratified-fluid flow may occurs in the ocean (owing to salinity variation) or in the atmosphere (owing to temperature variations). For 2D steady state stratified flow, the continuity equation should be

0)()(=

∂∂

+∂

∂yv

xu ρρ

Since the control volume is fixed in space, the time derivative can be placed inside the volume integral

Apply divergence theorem:

Integral form of continuity equation becomes:

Integral form continuity equation:

Differential form of continuity equation:

Steady and Unsteady flows

Unsteady:

Steady:

Compressible and Incompressible flows

Compressible:

Incompressible: constant

Summary of Continuity Equation

0)( =ρDtD

OUTLINE FOR Chapter 2-2


Basic Laws

• Conservation of mass – Continuity Equation• Conservation of (angular) Momentum

- Newton’s second law of motion.• Conservation of Energy

The first law of thermodynamics

Analysis of any problem in fluid mechanics necessarily includes statement of the basic laws governing the fluid motion. The basic laws, which applicable to any fluid, are:

BASIC LAWS FOR A SYSTEM- Conservation of Momentum

• Newton’s Second Law Stating 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 P is the linear momentum of the system

∫∫∫==

=

systemsystem

VdVDtD

DtPD

dtPdF ρ

• The forces act on fluid particles:– Body forces ( gravity, electromagnetic ).

– Surface forces ( pressure, viscous ).

Total viscous forces =

Review of Momentum Equation (I)


Momentum Equation

Physical principle:

Force = time rate of change of momentum

Body farces: gravity, electromagnetic forces, or anyother forces which “act at a distance on the fluidinside V.

Force:

Surface forces: pressure and shear stress acting onthe control surface S.

Total viscous forces =

Time rate of change of momentum

The integral form of momentum equations


Divergence theorem

The differential form of momentum equations (Navier-Stokes equations)

For unsteady 3D flow, compressible or incompressible, viscous or inviscid

For

Euler equations for steady inviscid flow

Inviscid, Incompressible: constant

xPuVVu∂∂

−=∇•+•∇ρ1)(

yPvVVv∂∂

−=∇•+•∇ρ1)(

zPwVVw∂∂

−=∇•+•∇ρ1)(

ρ

ρ

ρ-1

-1

-1

Review of Momentum Equation (II)

)()( 2

2

2

2

yu

xuf

xP

yuv

xuu

tu

x ∂∂

+∂∂

++∂∂

−=∂∂

+∂∂

+∂∂ µρρ

xx Fdxdzyudydz

xudxdydzfdPdydz

DtDudxdydzma =

∂∂

+∂∂

++−== )()(ρµρρ

dx

dzdy

12 PPdP −=

2P1P

)(

orce

dxdzyudydz

xuF

dxdzdydzfshear

visoucs

yxxx

∂∂

+∂∂

=

+=

µ

ττ

)()()( 2

2

2

2

yu

xuf

dxdP

dxdydzdxdz

yu

dxdydzdydz

xuf

dxdydzdPdydz

DtDu

xx ∂∂

+∂∂

++−=∂∂

+∂∂

++−= µρµρρ

)()()( 2

2

2

2

yu

xuf

dxdP

yuv

xuu

tu

DtDu

x ∂∂

+∂∂

++−=∂∂

+∂∂

+∂∂

= µρρρ

ααα∇•+

∂∂

= utDt

D Material derivative

SUMMARY


For steady, incompressible, inviscid flow, no body force

Continuity equation:

Momentum equation:

xP

zuw

yuv

xuuuV

∂∂

−=∂∂

+∂∂

+∂∂

=∇•ρ1

yP

zvw

yvv

xvuvV

∂∂

−=∂∂

+∂∂

+∂∂

=∇•ρ1

zP

zww

ywv

xwuwV

∂∂

−=∂∂

+∂∂

+∂∂

=∇•ρ1

0=∂∂

+∂∂

+∂∂

=•∇zw

yv

xuV

kwjviuzyxV

++=),,(

Vorticity• Defining Vorticity ζ which is a measurement of the

rotation of a fluid element as it moves in the flow field:

• In cylindrical coordinates system:

Vyu

xvk

xw

zuj

zv

ywi

×∇=

∂∂

−∂∂

+

∂∂

−∂∂

+

∂∂

−∂∂

== ωζ 2

∂∂

∂∂

∂∂

=×∇===

wvuzyx

kji

VVcurl

ωζ 2

∂∂

−∂∂

+

∂∂

−∂∂

+

∂∂

−∂∂

=×∇θθ

θθ

θ rz

zrzr

Vrr

rVr

er

Vz

Vez

VVr

eV 111

Fluid Rotation

∂∂

−∂∂

=

∂∂

−∂∂

=

∂∂

−∂∂

=

yu

xv

xw

zu

zv

yw

Z

y

x

212121

ω

ω

ω

( ) ( ) ( )[ ]

∂∂

−∂∂

+

∂∂

−∂∂

+

∂∂

−∂∂

=++==×∇=yu

xvk

xw

zuj

zv

ywikjiV zyx

((22 ωωωωζ

= 0 Irrotational≠ 0 rotational

Rotational flow Irrotational flow

0=×∇= VVcurl

0≠×∇= VVcurl

Irrotational flow rotational flow

Irrotational and Rotational flows

W3 8

EXAMPLE OF VORTICITY

( )2 ( )2

CIRCULATION

From Stokes’ theorem

Definition:

Irrotational flow

C

Vds

V∞

Circulation Lift

Example: Circulation in a uniform flow

u = V∞

v = 0

Irrotational flow

For arbitrary close curve C

(irrotational)

Stokes’ theorem


STREAMFUNCTION AND VELOCITY POTENTIAL

Streamfunction Ψ: definition

streamfunction Ψ properties:

automatically satisfy continuity equation

Velocity Potential φ: definition

Velocity potential φ properties:

automatically satisfy irrotational conditon

1.

2. Relationship between streamfunction and velocity potential

= 0

SUMMARY

substantial derivative

local derivative

convective derivative

= 0 Irrotational≠ 0 rotational

1. Substantial derivative

2. Streamline

3. Vorticity

4. Circulation

Irrotational flow

5. Streamfunction Ψ:

6. Velocity Potential φ:

OUTLINE FOR Chapter 2

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