Fluid Engineering Mechanics - جامعة نزوى · 2017-03-05 · Fluid Kinematics 2 Branch of...

Chapter 4

Fluid Kinematics: Steady and unsteady flow, laminar

and turbulent flow, uniform and non-uniform flow.

Path-line, streamlines and stream tubes. Velocity and

discharge. Control volume, Equation of continuity for

compressible and incompressible fluids.

Dr. Muhammad Ashraf Javid

Assistant Professor

1

Fluid Engineering Mechanics

Fluid Kinematics

2

Branch of fluid mechanics which deals with response of fluids in motion without considering forces and energies in them.

The study of kinematics is often referred to as the geometry of motion.

Car surface pressure contours

and streamlines

Flow around cylindrical object

Fluid Flow

3

Rate of flow: Quantity of fluid passing through any

section in a unit time.

Type:

1. Volume flow rate:

2. Mass flow rate

3. Weigh flow rate

time

fluid ofQuantity flow of Rate

time

fluid of volume

time

fluid of mass

time

fluid ofweight

Fluid Flow

4

Let’s consider a pipe in which a fluid is flowing with mean velocity, V (=L/t).

Let, in unit time, t, volume of fluid (AL) passes through section X-X,

1. Volume flow rate:

2. Mass flow rate:

3. Weigh flow rate:

V

L

A

Longitudinal Section Cross Section

AVt

ALQ

time

fluid of volume

QAV

t

ALM

time

fluid of mass

QAVG

t

AL

t

ALgG

time

fluid ofweight

Units

X

X

m3/sec

Kg/sec

N/sec

Reynolds Number (NO.)

5

It is the ratio between inertial forces and viscous

forces.

v

ieN

F

F

forceViscous

forceInertialRorR

1)(....... 2222

2

3 VLT

LL

T

LLavaMFi

2)()().(.2

2 VLT

LL

T

LL

LT

LA

dy

duAFv

VLVL

VL

VL

F

FRorR

v

ieN

22

L = characteristics length

L = D in case of pipe

L = R = hydraulic mean depth in case of open channel

It is used to differentiate between laminar and turbulent flow (nature of flow).

Types of Flow

6

Depending upon fluid properties

Ideal and Real flow

Incompressible and compressible flow

Depending upon properties of flow

Laminar and turbulent flows

Steady and unsteady flow

Uniform and Non-uniform flow

Rotational flow and Ir-rotational flow

Critical flow, Sub critical flow, and Super critical flow

Ideal and Real flow, and Velocity Profile

7

Real fluid flows implies friction effects. Ideal fluid flow is hypothetical;

it assumes no friction.

Velocity distribution of pipe flow

Compressible and incompressible flows

8

Incompressible fluid flows assumes the fluid have constant density

while in compressible fluid flows density is variable and becomes

function of temperature and pressure.

P1 P2

v1

v2

v2

P1

P2

v1

v2

Incompressible fluid Compressible fluid

Laminar and turbulent flow

9

The flow in laminations (layers) is

termed as laminar flow while the

case when fluid flow layers

intermix with each other is termed

as turbulent flow.

Transition of flow from Laminar to turbulent

Laminar flow

Turbulent flow

Laminar and turbulent flow

10

• Re ≤ 2000, flow is laminar in pipe and Re > 4000, flow in turbulent

• Re ≤ 500, flow is laminar in open channel, Re > 1000, flow is turbulent

• In-between of above values, the flow is in transition stage.

Steady and Unsteady flows

11

Steady flow: It is the flow in

which conditions of flow remains

constant w.r.t. time at a particular

section but the condition may be

different at different sections.

Flow conditions: velocity, pressure,

density or cross-sectional area etc.

e.g., constant discharge through a

pipe

Unsteady flow: It is the flow in

which conditions of flow changes

w.r.t. time at a particular section.

e.g., variable discharge through a

pipe

V

Longitudinal Section

X

X

conttVdt

d

dt

dp

dt

dv ;0,0,0

variable;0,0,0 Vdt

d

dt

dp

dt

dv

Uniform and Non-uniform flow

12

Uniform flow: It is the flow in which conditions of flow remains constant from section to section at any instant of time.

e.g., Constant discharge though a constant diameter pipe

Non-uniform flow: It is the flow in which conditions of flow does not remain constant from section to section.

e.g., Constant discharge through variable diameter pipe

V

Longitudinal Section

X’

X’

V

Longitudinal Section X’

conttVdt

d

dt

dp

dt

dv ;0,0,0

variable0,0,0 Vdt

d

dt

dp

dt

dv

X’

X

X

X

X

Describe flow condition

13

Constant discharge though non

variable diameter pipe

V

Longitudinal Section X

Steady flow !!

Non-uniform flow !!

X

Steady-non-uniform flow

13

Variable discharge though non

variable diameter pipe

V

Longitudinal Section X

Unsteady flow !!

Non-uniform flow !!

X

unsteady-non-uniform flow

Flow Combinations

14

Type Example

1. Steady Uniform flow Flow at constant rate through a duct

of uniform cross-section

2. Steady non-uniform flow

Flow at constant rate through a duct

of non-uniform cross-section (tapering

pipe)

3. Unsteady Uniform flow Flow at varying rates through a long

straight pipe of uniform cross-section.

4. Unsteady non-uniform flow Flow at varying rates through a duct

of non-uniform cross-section.

Other Flow Types

Rotational Flow In this case the fluid particles are moving around a curved path about a

fixed axis of rotation.

Ir-rotational Flow Fluid particles are not moving around a curved path about a fixed axis of

rotation.

Critical Flow Velocity of flow becomes equal to critical velocity.

Critical velocity is the velocity below which all the turbulence will be damaged out by the viscosity of fluid. V = Vcr , D = Dcr

Sub critical Flow V < Vcr and Depth of flow > critical depth of flow

Super critical Flow V > Vcr and Depth of flow < critical depth of flow

Froude Number and Flow States

16

The Froude number, Fr, is a dimensionless value that describes different flow regimes of open channel flow. The Froude number is a ratio of inertial and gravitational forces.

Gravity (numerator) - moves water downhill

Inertia (denominator) - reflects its willingness to do so.

Where:

V = Water velocity

D = Hydraulic depth (cross sectional area of flow / top width)

g = Gravity

When:

Fr = 1, critical flow,

Fr > 1, supercritical flow (fast rapid flow),

Fr < 1, subcritical flow (slow / tranquil flow)

gD

VFr

One, Two and Three Dimensional Flows

17

Although in general all fluids flow three-dimensionally, with

pressures and velocities and other flow properties varying in all

directions, in many cases the greatest changes only occur in two

directions or even only in one. In these cases changes in the other

direction can be effectively ignored making analysis much more

simple.

Flow is one dimensional if the flow parameters (such as velocity,

pressure, depth etc.) at a given instant in time only vary in the

direction of flow and not across the cross-section

Longitudinal section of rectangular channel Cross-section Velocity profile

Mean

velocity Water surface

One, Two and Three Dimensional Flows

18

Flow is two-dimensional if it can be

assumed that the flow parameters

vary in the direction of flow and in

one direction at right angles to this

direction

Flow is three-dimensional if the flow

parameters vary in all three directions

of flow

Two-dimensional flow over a weir

Three-dimensional flow in stilling basin

Visualization of flow Pattern

19

Streamlines around a wing shaped body Flow around a skiing athlete

Path line and stream line

20

Path-line: It is a trace made by

single particle over a period of time.

Streamline is a imaginary line

shows the mean direction of a

number of particles at the same

instance of time.

Character of Streamline

Streamlines can not cross each other.

Streamline can't be a folding line, but

a smooth curve.

Streamline cluster density reflects

the magnitude of velocity.

(Dense streamlines mean large velocity;

while sparse streamlines mean small

velocity.） Flow around cylindrical object

Mean Velocity and Discharge

21

Let’s consider a fluid flowing with mean velocity, V, in a pipe of uniform cross-

section, A,. Thus volume of fluid that passes through section XX in unit time ,

Δt, becomes;

Volume flow rate:

V

V Δt

A

Longitudinal Section Cross Section

AV

t

AtVQ

time

fluid of volume

X

X

AtVfluid of Volume

VAG

VAM

Similarly

Velocity

Distribution

Vmax

V=Vavg

Continuity

22

In a continuous flowing fluid, quantity of fluid flowing per unit

time is same for each section.

Assumptions:

Fluid is ideal (viscosity is zero).

Stream surface is ideal (frictionless).

Flow is steady.

Law of conservation of mass is applicable.

Continuity

23

Matter cannot be created or destroyed

- (it is simply changed in to a different

form of matter).

This principle is know as the

conservation of mass and we use it in the

analysis of flowing fluids.

The principle is applied to fixed

volumes, known as control volumes

shown in figure:

An arbitrarily shaped control volume.

For any control volume the principle of conservation of mass says

Mass entering per unit time - Mass leaving per unit time

= Increase of mass in the control volume per unit time

Continuity Equation

24

For steady flow there is no increase in the mass within the control

volume, so

Mass entering per unit time = Mass leaving per unit time

A stream tube

Derivation:

Lets consider a stream tube.

ρ1, v1 and A1 are mass density, velocity and cross-sectional area at section 1. Similarly, ρ2, v2 and A2 are mass density, velocity and cross-sectional area at section 2.

According to mass conservation

2222

1111

VAM

VAM

dt

MdVAVA

dt

MdMM

CV

CV

222111

21

Continuity Equation

25

For steady flow condition

Hence, for stead flow condition, mass flow rate at section 1= mass flow rate at section 2. i.e., mass flow rate is constant.

Similarly

Assuming incompressible fluid,

Therefore, according to mass conservation for steady flow of incompressible fluids volume flow rate remains same from section to section.

0/ dtMd CV

222111222111 0 VAVAVAVA

222111 VAVAM

222111 VgAVgAG

21

2211 VAVA 21 QQ 4321 QQQQ

Continuity Equation

26

For incompressible fluids:

Q=A1V1=A2v2=A3v3= constant

For compressible fluids:

M=ρ1A1v1=ρ2A2v2=ρ3A3v3= constant

or

W=γ1A1v1=γ2A2v2=γ3A3v3= constant

NUMERICALS

27

If the diameter at section 1 is d1 = 30mm and at section 2 d2 = 40mm and

the mean velocity at section 2 is u2 = 3.0 m/s, determine velocity at Section

1.

NUMERICALS

28

If pipe 1 diameter = 50mm, mean velocity 2m/s,

pipe 2 diameter 40mm takes 30% of total

discharge and pipe 3 diameter 60mm. What are

the values of discharge and mean velocity in

each pipe?

29

Fluid Dynamics: Hydrodynamics: Different forms of energy in a

flowing liquid, head, Bernoulli's equation and its

application, Energy line and Hydraulic Gradient Line,

and Energy Equation, Free and forced vortex, Forces

on pressure conduits, reducers and bends

Hydrodynamics

It is the study of liquids in motion

considering the forces and energies

exerted by or upon the liquid.

Forms of Energy

Liquid has three form of energy.

Kinetic Energy

If a liquid mass “m” is flowing with velocity “V” and all the liquid

particles are moving with same velocity then the K.E. = (1/2)mV2

Pressure energy: If a liquid exist under a pressure “p” then the Pr.E. of that liquid will

be.

Pr.E = p.dv

dv = small volume of liquid

V

Z

Q

Datum line

Forms of Energy

Potential Energy:

If a liquid particles of weight “w” is situated at a elevation “z” above datum line then P.E

P.E = WZ = mg z

The P.E of a particle of the liquid depends on its elevation above any arbitrary datum.

We are only interested in difference of elevation and therefore the location of datum is determined solely by consideration of convenience.

1

2

Z1

Z2

W Z1 W Z2

Types of head

p

W

dvpheadPressure

g

V

mg

Vm

W

VmheadKinetic

zW

ZWheadPotential

.

222

222

All the three heads have the unit of length (m).

Head: energy per unit weight is called as head. = E/W • For all three forms of energy we have three different head;

• Kinetic head, • potential head or elevation head, • pressure head.

• When elevation head is zero, the pressure head is also called as static head, otherwise static head is equal to pressure head plus elevation head.

Bernoulli's Theorem

For a perfect, frictionless,

incompressible liquid the total

head of the particles of liquid

moving in a continuous stream

remains constant.

It is another form of law of

conservation of energy.

P2

D

P1

Z1

Z2

Q

A

B

C

A’

B’ C’

D’

1

2

ds1

ds2

Assumptions

Fluid is ideal

Fluid is incompressible

Flow is steady

There is no loss of

energy due to friction

Bernoulli's Theorem

Consider a perfect, incompressible, liquid is flowing through a pipe.

Consider a controlled volume of liquid between section-1 and section-2.

Original position of the volume of liquid can be defined by position A, B, C, D.

Let V1 is velocity of flow, Z1 elevation, p1 pressure at section-1 respectively.

Similarly V2 is velocity of flow, Z2 elevation, p2 pressure at section-2

respectively.

Let us assume this volume displaces small distance ds1 at section-1 due to

pressure p1.

Then as the fluid is incompressible, so it will also move by a distance ds2 at

section-2.

Such that the new position of the liquid is defined as A`, B`, C`, D`.

Bernoulli's Theorem

)(2

)( 21

2

1

2

221 ZZWVVg

Wppdv

)1(.. EPinlossEKingainpressurebydonework

Wdvandppdvpressurebydonework )2()( 21

)3()(2

. 2

1

2

2 VVg

WEKinGain

)4()(. 21 ZZWEPinLoss

)5()(2

)( 21

2

1

2

221 ZZWVVg

Wpp

W

Put equation 2, 3 and 4 in equation 1

Bernoulli's Theorem

Dividing the previous equation (5) by W

22

2

2

2

11

1

22Z

p

g

V

g

VZ

p

)(2

1)(

121

2

1

2

221 ZZVVg

pp

21

2

1

2

221

22ZZ

g

V

g

Vpp

)6(22

2

2

22

1

2

11

p

g

VZ

p

g

VZ

Energy Equation

This equation (5) is applicable only for 1st three

assumptions.

Considering the last assumption of Bernoulli’s Theorem

hL is the loss of head due to friction.

The above equation is called as Energy Equation:

Lhp

g

VZ

p

g

VZ

2

2

22

1

2

11

22

Application of Bernoulli's Principle

Operation of an airplane wing

Design of Venturimeter •The shape of the wing forces the

air moving above the wing to move

faster than the air moving below

the wing.

•This cause the pressure above the

wing to be lower than the pressure

below the wing.

•The pressure difference pushes up

on the wing.

•This is called as lift.

Terminologies

Static head

Sum of pressure and elevation head.

Static head = Z + (p/γ)

If datum is selected in such a way that elevation of the fluid

particles is zero then static head will be equal to pressure

head only.

Terminologies

Hydraulic grade line (H.G.L)

It is the graphical representation of the static head in the form of a line above any datum along the flow path.

Static head = Z + (p/γ)

The height of piezometer tube indicates the static head.

Every ordinate of H.G.L. gives us the value of static head w.r.t. the relative datum.

If we install piezometer at different section then the line joining the piezometeric liquid level will be the H.G.L. that is why H.G.L. is also called as piezometer head line.

Q

H.G.L.

Terminologies

Energy Line

Total energy head

Total energy head = elevation head + pressure head + velocity head

If there is no head loss then the total energy head will be equal to total head.

Energy Grade Line (E.G.L.)

Graphical representation of the total energy head in the form of line above any datum along the flow path.

Every ordinate of E.L gives us the value of total energy head with respect to the datum.

If we install pitot tube at a section then the level above datum to which liquid will rise in the tube reflect the value of total energy head and the line joining these levels will be the energy line.

As liquid entering into tube will comes into rest and transfer all kinetic energy into potential energy.

p

Datum line

Z Z

p

g

V

2

2

Examples to H.G.L. and E.G.L.

p

g

V

2

2

Datum line Z Z

Datum line Z Z

•hL= o

•Velocity and

pressure head is

same

•hL≠ o

•Velocity head is

same and pressure

head will change

E.G.L.

H.G.L

hL

g

V

2

2

E.G.L.

H.G.L

p


•hL= o

•Velocity and

pressure head is

changing.

p

g

V

2

2

E.G.L

H.G.L

p

p

g

V

2

2

E.G.L

H.G.L

p

hL •hL≠ o

•Velocity and

pressure head is

changing.

Problem

49

5.2.3

Apply Bernoulli's Equation

Problem

50

Power

51

Rate of work done is termed as power

Power=Energy/time

Power=(Energy/weight)(weight/time)

If H is total head=total energy/weight and γQ is the weight flow rate

then above equation can be written as

Power=(H)(γQ)= γQH

In BG:

Power in (horsepower)=(H)(γQ)/550

In SI:

Power in (Kilowatts)=(H)(γQ)/1000

1 horsepower=550ft.lb/s

Problem

52

53

Free and Forced Vortex Flow Momentum and Forces in Fluid Flow

Free and Forced Vortex Flow

54

Vortex flow is defined as flow along curved path. Or It is a type of

flow in which the liquid particles are continuously moving

around a curved path about any axis of rotation.

It is of two types namely; (1). Free vortex flow and (2) forced

vortex flow

If the fluid particles are moving around a curved path with the help

of some external torque the flow is called forced vortex flow. And if

no external force is acquired to rotate the fluid particle, the flow is

called free vortex flow.

Forced Vortex Flow (Rotational Flow)

55

It is defined as that type of flow, in which some external torque is

required to rotate the fluid mass.

The fluid mass in this type of flow rotate at constant angular

velocity, ω. The tangential velocity, V, of any fluid particle is given by

V= ω r,

Where, r is radius of fluid particle from the axis of rotation

Examples of forced vortex flow are;

1. A vertical cylinder containing liquid which is rotated about its central axis

with a constant angular velocity ω,

2. Flow of liquid inside impeller of a centrifugal pump

3. Flow of water through runner

Forced Vortex Flow (Rotational Flow)

56

Free Vortex Flow (Irrotational flow)

57

When no external torque is required to rotate the fluid mass, that

type of flow is called free vortex flow.

Thus the liquid in case of free vortex flow is rotating due to the

rotation which is imparted to the fluid previously.

Example of free vortex flow are

1. Flow of liquid through a hole provided at the bottom of container

2. Flow of liquid around a circular bend in pipe

3. Flow of fluid in a centrifugal pump casing

Forced vortex

58

It is a type of flow in which

particles of liquid are forced to

move in a curved path about any

axis of rotation.

In this case fluid is subjected to

external torque.

We are only interested to profile

of water surface and level (y).

Consider a cylinder rotated at ω

(rad/sec). ω(rad/sec).

Forced vortex

59

The free surface of water will

not remain level but get a curve

shape as shown in the figure.

Consider a point “P” on the

free surface of forced vortex.

Let

W = weight of particle

F = centrifugal force

R = reaction of F and W

making “θ” with vertical.

Forced vortex

60

x = radius of the point with

respect to axis of rotation.

r = radius of cylinder

V = Tangential velocity

xVxrFor

rV

,

r

VmF

2

xg

W

x

x

g

WF 2

22

tan

sin

)2()1(

)2(

)1(sin

sin

2

2

2

g

x

CosR

R

W

xg

W

byEqdividing

CosRW

oFy

Rxg

W

RF

oFx

Forced vortex

61

)4(2

,

)3(2

xw.r.t.equationabovegintegratin

tan

22

22

2

g

rY

Yyrxwhen

g

xy

dx

dy

g

x

dx

dy

Equation (3) states that

free surface of liquid

subjected to forced

vortex will by paraboloid.

Height of paraboloid w.r.t. original surface

62

Height of paraboloid w.r.t. original surface

63

Consider “AA” level surface when

there is no rotation.

“BB” is the lowest level when

cylinder is subjected to rotation

and “CC” is maximum level of

paraboloid.

Volume of liquid between “BB”

and “CC” will remain same as the

volume between “AA” and “BB”

when there is no rotation.

Volume of liquid before rotation =

volume of liquid after rotation

)5(2

)2

(

2

22

222

Yh

YYrhr

YrYrhr

•Equation (5) states that

depression at the axis of

rotation (or at center) w.r.t

AA is same as elevation at

radius r.

Free vortex

64

It is a type of flow in

which particles of a

liquid move in a curve

path about any axis of

rotation without any

external force.

Consider a particle of

the liquid at the surface

of free vortex.

Free vortex

65

Let

m = mass of particle

v = tangential velocity

r = radius of circular path

Angular momentum of this particle is

L = M.r = mvr

As torque is defined as rate of change of angular momentum

In case of free vortex the product of tangential velocity and corresponding radius is constant.

2211

1

1

timew.r.titgintegratin

)(0

0

)(

rvrvC

vrm

C

mvrC

dt

mvrd

T

dt

mvrdT

Numerical: Forced vortex flow

66

Y

Y

Numerical: Forced Vortex flow

67

Y

Y

= h

= h

Numerical: Free Vortex Flow

68

Numerical: Free Vortex Flow

69

Momentum and Forces in Fluid Flow

70

We have all seen moving fluids exerting forces. The lift force on an aircraft

is exerted by the air moving over the wing. A jet of water from a hose

exerts a force on whatever it hits.

In fluid mechanics the analysis of motion is performed in the same way as in

solid mechanics - by use of Newton’s laws of motion.

i.e., F = ma, which is used in the analysis of solid mechanics to relate applied

force to acceleration.

In fluid mechanics it is not clear what mass of moving fluid we should use,

so, we use a different form of the equation.

dt

mdma sV

F


71

Newton’s 2nd Law can be written:

The Rate of change of momentum of a body is equal to the resultant force acting

on the body, and takes place in the direction of the force.

The symbols F and V represent vectors and so the change in momentum must be

in the same direction as force.

It is also termed as impulse momentum principle

dt

md sVF

mV

F Sum of all external forces on a body of fluid or systems

Momentum of fluid body in direction s

smddt VF


72

Let’s start by assuming that we

have steady flow which is non-uniform

flowing in a stream tube.

In time δt a volume of the fluid

moves from the inlet a distance u1 δt ,

so the volume entering the stream-

tube in the time δt is A stream-tube in three and two-dimensions

volume entering the stream tube = area x distance

mass entering stream tube = volume x density

momentum of fluid entering stream tube = mass x velocity

tuA 11

tuA 111

1111 utuA

momentum of fluid leaving stream tube 2222 utuA


73

Now, according to Newton’s 2nd Law the force exerted by the fluid

is equal to the rate of change of momentum. So

Force=rate of change of momentum

We know from continuity of incompressible flow, ρ=ρ1= ρ2 &

Q=Q1=Q2

111222

11112222

1111222211112222

F

F

uQuQt

utuA

t

utuA

t

tuuA

t

tuuA

t

tuuAtuuA

1212 uumuuQF

This analysis assumed that the inlet and outlet velocities were in the

same direction - i.e. a one dimensional system. What happens when

this is not the case?


74

Consider the two dimensional

system in the figure below:

At the inlet the velocity vector, u1 ,

makes an angle, θ1 , with the x-axis,

while at the outlet u2 make an

angle θ 2.

In this case we consider the forces

by resolving in the directions of the

co-ordinate axes.

The force in the x-direction

Two dimensional flow in a streamtube


75

The force in the y-direction

The resultant force can be determined by combining Fx and Fy

vectorially as

And the angle at which F acts is given by


76

For a three-dimensional (x, y, z) system we then have an extra force

to calculate and resolve in the z direction.

This is considered in exactly the same way.

In summary we can say:

The total force of the fluid = rate of change of momentum through the control volume

Fluid Engineering Mechanics - جامعة نزوى · 2017-03-05 · Fluid Kinematics 2 Branch of...

Documents

Transcript of Fluid Engineering Mechanics - جامعة نزوى · 2017-03-05 · Fluid Kinematics 2 Branch of...