fluid 06

1

FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 6 Flow Analysis Chapter 6 Flow Analysis Using Differential MethodsUsing Differential Methods

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

National Taiwan UniversityNational Taiwan University12/24/200712/24/2007

2

MAIN TOPICSMAIN TOPICS

Fluid Element KinematicsFluid Element KinematicsConservation of MassConservation of MassConservation of Linear MomentumConservation of Linear MomentumInviscidInviscid FlowFlowSome Basic, Plane Potential FlowSome Basic, Plane Potential Flow

3

Motion of a Fluid ElementMotion of a Fluid Element

Fluid Translation: The element moves from one point to another.Fluid Translation: The element moves from one point to another.Fluid Rotation: The element rotates about any or all of the x,y,Fluid Rotation: The element rotates about any or all of the x,y,z axesz axes..Fluid Deformation:Fluid Deformation:

Angular Deformation:The elementAngular Deformation:The element’’s angles between the sides s angles between the sides change.change.Linear Deformation:The elementLinear Deformation:The element’’s sides stretch or contract.s sides stretch or contract.

4

Fluid Translation Fluid Translation velocity and accelerationvelocity and acceleration

The velocity of a fluid The velocity of a fluid particle particle can be expressedcan be expressed

The The total accelerationtotal acceleration of the particle is given byof the particle is given bykwjviu)t,z,y,x(VVrrrrv

++==

zVw

yVv

xVu

tV

DtVDa

wdtdz,v

dtdy,u

dtdx

dtdz

zV

dtdy

yV

dtdx

xV

tV

DtVDa

∂∂

+∂∂

+∂∂

+∂∂

==⇒

===

∂∂

+∂∂

+∂∂

+∂∂

==

rrrrrr

rrrrrr

tDVDar

r= is called the material , or substantial derivative.is called the material , or substantial derivative.

Acceleration field

Velocity field

5

Physical SignificancePhysical Significance

tV

zVw

yVv

xVu

tDVDa

∂∂

+∂∂

+∂∂

+∂∂

==rrrrr

r

TotalAcceleration Of a particle Convective

Acceleration

LocalLocalAccelerationAcceleration

tVV)V(

tDVDa

∂∂

+∇⋅==r

rvr

r

6

Scalar ComponentScalar Component

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

∂∂

+∂∂

+∂∂

+∂∂

=

Rectangular Rectangular coordinates systemcoordinates system

zVVV

rV

rVV

tVa

zVV

rVVV

rV

rVV

tVa

zVV

rVV

rV

rVV

tVa

zz

zzr

zz

zr

r

rz

2rr

rr

r

∂∂

+θ∂

∂+

∂∂

+∂∂

=

∂∂

++θ∂

∂+

∂∂

+∂∂

=

∂∂

+−θ∂

∂+

∂∂

+∂∂

=

θ

θθθθθθθ

θθ

Cylindrical Cylindrical coordinates systemcoordinates system

7

Linear TranslationLinear Translation

All points in the element have All points in the element have the same velocity (which is only the same velocity (which is only true if there are o velocity true if there are o velocity gradients), then the element will gradients), then the element will simply translate from one simply translate from one position to another.position to another.

8

Linear Deformation Linear Deformation 1/21/2

The shape of the fluid element, described by the angles at The shape of the fluid element, described by the angles at its vertices, remains unchanged, since its vertices, remains unchanged, since all right angles all right angles continue to be right anglescontinue to be right angles..A change in the x dimension requires a nonzero value of A change in the x dimension requires a nonzero value of

A A ……………………………… y y A A ……………………………… z z z/w ∂∂

y/v ∂∂x/u ∂∂

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Linear Deformation Linear Deformation 2/22/2

The change in length of the sides may produce change in The change in length of the sides may produce change in volume of the element.volume of the element.

The change inThe change in )t)(zy(xxuV δδδ⎟

⎠⎞

⎜⎝⎛ δ∂∂

=δ

The rate at which the The rate at which the δδV is changing per V is changing per unit volume due to gradient unit volume due to gradient ∂∂u/ u/ ∂∂xx

( )xu

dtVd

V1

∂∂

=δ

δIf If ∂∂v/ v/ ∂∂y and y and ∂∂w/ w/ ∂∂z are involvedz are involved

Volumetric dilatation rateVolumetric dilatation rate( ) V

zw

yv

xu

dtVd

V1 r

⋅∇=∂∂

+∂∂

+∂∂

=δ

δ

10

Angular Rotation Angular Rotation 1/41/4

δtδαω lim

0δtOA

→

=The angular velocity of line OA

txv

x

txxv

tan δ∂∂

=δ

δδ⎟⎠⎞

⎜⎝⎛∂∂

=δα=δα &For small angles

xv

OA ∂∂

=ω

yu

OB ∂∂

=ω

CWCW

CCWCCW

“-” for CCW

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( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω+ω=ωyu

xv

21

21

OBOAz

The rotation of the element about the zThe rotation of the element about the z--axis is defined as the axis is defined as the average of the angular velocities average of the angular velocities ωωOAOA and and ωωOBOB of the two of the two mutually perpendicular lines OA and OB.mutually perpendicular lines OA and OB.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ωzv

yw

21

x ⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=xw

zu

21

yω

kji zyxrvvr

ω+ω+ω=ωIn vector formIn vector form

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⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω

⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=ω⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω

kyu

xvj

xw

zui

zv

yw

21

xw

zu

21

zv

yw

21

yx

rrrr

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ωyu

xv

21

z

kyu

xv

21j

xw

zu

21i

zv

yw

21V

21Vcurl

21 rrrrrr

⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

+⎥⎦⎤

⎢⎣⎡

∂∂

−∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

=×∇==ω

Defining Defining vorticityvorticity V2rr

×∇=ω=ζ

Defining Defining irrotationirrotation 0V =×∇r

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kyu

xv

21j

xw

zu

21i

zv

yw

21

wvuzyx

kji

21V

21Vcurl

21

rrr

rrr

rrr

⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

+⎥⎦⎤

⎢⎣⎡

∂∂

−∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

=

∂∂

∂∂

∂∂

=×∇==ω

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VorticityVorticity

Defining Defining VorticityVorticity ζζ whichwhich is a measurement of the rotation of a is a measurement of the rotation of a fluid elementfluid element as it moves in the flow field:as it moves in the flow field:

In cylindrical coordinates systemIn cylindrical coordinates system::

V21k

yu

xvj

xw

zui

zv

yw

21 rrrrr

×∇=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω

VVcurl2rrrr

×∇==ω=ζ

⎟⎠⎞

⎜⎝⎛

θ∂∂

−∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

+⎟⎠⎞

⎜⎝⎛

∂∂

−θ∂

∂=×∇ θ

θθ r

zzrz

rV

r1

rrV

r1e

rV

zVe

zVV

r1eV

rrrr

15

Angular Deformation Angular Deformation 1/21/2

Angular deformation of a particle is given by the sum of the twoAngular deformation of a particle is given by the sum of the twoangular deformationangular deformation

δβ+δα=δγ

txxvtvtx

xvvty

yututy

yuu δδ

∂∂

=δ−δ⎟⎠⎞

⎜⎝⎛ δ

∂∂

+=δηδδ∂∂

=δ−δ⎟⎟⎠

⎞⎜⎜⎝

⎛δ

∂∂

+=δξ

y/x/ δδξ=δβδδη=δα

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

==δ

⎟⎟⎠

⎞⎜⎝

⎛δ

δδ

∂∂+δ

δδ

∂∂

=δδβ+δα

=γ→δ→δ y

uxv...

t

tyy

yut

xx

xv

limt

lim0t0t

&

ξ（Xi）η（Eta）

Rate of shearing strain or the rate of angular deformationRate of shearing strain or the rate of angular deformation

16

Angular Deformation Angular Deformation 2/22/2

The rate of angular deformation in xy plane

The rate of angular deformation in yz plane

The rate of angular deformation in zx plane

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

yu

xv

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

zv

yw

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

zu

xw

17

Example 6.1 Example 6.1 VorticityVorticity

For a certain twoFor a certain two--dimensional flow field dimensional flow field thth evelocityevelocity is given by is given by

Is this flow Is this flow irrotationalirrotational? ?

j)yx(2ixy4V 22rrr

−+=

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Example 6.1 Example 6.1 SolutionSolution

0wyxvxy4u 22 =−==

0xw

zu

21

0zv

yw

21

y

x

=⎟⎠⎞

⎜⎝⎛

∂∂

−∂∂

=ω

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω

0yu

xv

21

z =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

=ω

This flow is This flow is irrotationalirrotational

19

Conservation of Mass Conservation of Mass 1/51/5

With field representation, the property fields are defined With field representation, the property fields are defined by continuous functions of the space coordinates and time.by continuous functions of the space coordinates and time.To derive the differential equation for conservation of To derive the differential equation for conservation of mass in rectangular and in cylindrical coordinate system.mass in rectangular and in cylindrical coordinate system.The derivation is carried out by applying conservation of The derivation is carried out by applying conservation of mass to a differential control volume.mass to a differential control volume.

With the control volume representation of the conservation of maWith the control volume representation of the conservation of massss

0dAnVVdt CSCV =∫ ⋅+∫∂∂ rv

The differential form of continuity equation???The differential form of continuity equation???

20


The CV chosen is an infinitesimal cube with sides of length The CV chosen is an infinitesimal cube with sides of length δδx, x, δδ y, y, and and δδ z.z.

zyxt

Vdt CV δδδρρ

∂∂

=∂∂∫

( )2x

xuu|u

2dxx

δ∂ρ∂

+ρ=ρ⎟⎠⎞

⎜⎝⎛+

( )2x

xuu|u

2xx

δ∂ρ∂

−ρ=ρ⎟⎠⎞

⎜⎝⎛ δ−

Net rate of mass Outflow in x-direction

21


Net rate of mass Net rate of mass Outflow in xOutflow in x--directiondirection

( ) ( ) ( ) zyxxuzy

2x

xuuzy

2x

xuu δδδ

∂ρ∂

=δδ⎥⎦⎤

⎢⎣⎡ δ

∂ρ∂

−ρ−δδ⎥⎦⎤

⎢⎣⎡ δ

∂ρ∂

+ρ=

Net rate of mass Net rate of mass Outflow in yOutflow in y--directiondirection

( ) zyxyv

δδδ∂ρ∂

=⋅⋅⋅⋅⋅=

Net rate of mass Net rate of mass Outflow in zOutflow in z--directiondirection

( ) zyxzw

δδδ∂ρ∂

=⋅⋅⋅⋅⋅=

22


Net rate of mass Outflow

( ) ( ) ( ) zyxzw

yv

xu

δδδ⎥⎦

⎤⎢⎣

⎡∂ρ∂

+∂ρ∂

+∂ρ∂

The differential equation for conservation of massThe differential equation for conservation of mass

( ) ( ) ( ) 0Vtz

wyv

xu

t=ρ⋅∇+

∂ρ∂

=∂ρ∂

+∂ρ∂

+∂ρ∂

+∂ρ∂ r

Continuity equationContinuity equation

23


Incompressible fluidIncompressible fluid

Steady flowSteady flow

0Vzw

yv

xu

=⋅∇=∂∂

+∂∂

+∂∂ r

0Vz

)w(y

)v(x

)u(=ρ⋅∇=

∂ρ∂

+∂ρ∂

+∂ρ∂ r

24

Example 6.2 Continuity EquationExample 6.2 Continuity Equation

The velocity components for a certain incompressible, steady floThe velocity components for a certain incompressible, steady flow w field arefield are

Determine the form of the z component, w, required to satisDetermine the form of the z component, w, required to satisfy the fy the continuity equation.continuity equation.

?wzyzxyvzyxu 222

=++=++=

25

Example 6.2 Example 6.2 SolutionSolution0

zw

yv

xu

=∂∂

+∂∂

+∂∂The continuity equationThe continuity equation

)y,x(f2zxz3w

zx3)zx(x2zw

zxyv

z2xu

2

+−−=⇒

−−=+−−=∂∂

+=∂∂

=∂∂

26

Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 1/31/3

The CV chosen is an infinitesimal cube with sides of The CV chosen is an infinitesimal cube with sides of length length drdr, rd, rdθθ, and , and dzdz..The net rate of mass flux out through the control surfaceThe net rate of mass flux out through the control surface

zrzVrV

rVrV zr

r δθδδ⎥⎦⎤

⎢⎣⎡

∂ρ∂

+θ∂

ρ∂+

∂ρ∂

+ρ θ

The rate of change of mass The rate of change of mass inside the control volumeinside the control volume

drdzrdt

θρ∂∂

27

Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 2/32/3

The continuity equationThe continuity equation

By By ““DelDel”” operatoroperator

The continuity equation becomesThe continuity equation becomes

0z

)V()V(r1

r)Vr(

r1

tzr =

∂ρ∂

+θ∂

ρ∂+

∂ρ∂

+∂ρ∂ θ

zk

r1e

rer ∂

∂+

θ∂∂

+∂∂

=∇ θ

rrr

0Vt

=ρ⋅∇+∂ρ∂ r

28

Conservation of MassConservation of MassCylindrical Coordinate SystemCylindrical Coordinate System 3/33/3

Incompressible fluidIncompressible fluid

Steady flowSteady flow

0Vz

)V()V(r1

r)rV(

r1 zr =⋅∇=

∂∂

+θ∂

∂+

∂∂ θ

r

0Vz

)V()V(r1

r)Vr(

r1 zr =ρ⋅∇=

∂ρ∂

+θ∂

ρ∂+

∂ρ∂ θ

r

29

Stream Function Stream Function 1/61/6

Streamlines ?Streamlines ? Lines tangent to the instantaneous velocity vectors at Lines tangent to the instantaneous velocity vectors at every point.every point.Stream function Stream function ΨΨ(x,y)(x,y) [Psi] ? Used to represent the velocity [Psi] ? Used to represent the velocity component u(x,y,t) and v(x,y,t) of a twocomponent u(x,y,t) and v(x,y,t) of a two--dimensional dimensional incompressible flow.incompressible flow.Define a function Define a function ΨΨ(x,y), called the stream function, which relates (x,y), called the stream function, which relates the velocities shown by the figure in the margin asthe velocities shown by the figure in the margin as

xv

yu

∂ψ∂

−=∂ψ∂

=

30


The stream function The stream function ΨΨ(x,y) (x,y) satisfies the twosatisfies the two--dimensional form of dimensional form of the incompressible continuity equationthe incompressible continuity equation

ΨΨ(x,y) (x,y) ?? Still unknown for a particular problem, but at least we Still unknown for a particular problem, but at least we have simplify the analysis by having to determine only one have simplify the analysis by having to determine only one unknown, unknown, ΨΨ(x,y)(x,y) , rather than the two function u(x,y) and v(x,y)., rather than the two function u(x,y) and v(x,y).

0xyyx

0yv

xu 22

=∂∂ψ∂

−∂∂ψ∂

⇒=∂∂

+∂∂

31


Another advantage of using stream function is related to the facAnother advantage of using stream function is related to the fact that t that line along which line along which ΨΨ(x,y) =constant(x,y) =constant are streamlines.are streamlines.How to prove ? From the definition of the streamline that the slHow to prove ? From the definition of the streamline that the slope ope at any point along a streamline is given byat any point along a streamline is given by

uv

dxdy

streamline

=⎟⎠⎞

Velocity and velocity component along a streamlineVelocity and velocity component along a streamline

32


The change of The change of ΨΨ(x,y) as we move from one point (x,y) (x,y) as we move from one point (x,y) to a nearly point (to a nearly point (x+dx,y+dyx+dx,y+dy) is given by) is given by

0udyvdx0d

udyvdxdyy

dxx

d

=+−>>=ψ>>

+−=∂ψ∂

+∂ψ∂

=ψ

uv

dxdy

streamline

=⎟⎠⎞

Along a line of constant Along a line of constant ΨΨ

This is the definition for a streamline. Thus, if we know the fuThis is the definition for a streamline. Thus, if we know the function nction ΨΨ(x,y) we can (x,y) we can plot lines of constant plot lines of constant ΨΨto provide the family of streamlines that are helpful in to provide the family of streamlines that are helpful in visualizing the pattern of flow. There are an infinite number ofvisualizing the pattern of flow. There are an infinite number of streamlines that make up streamlines that make up a particular flow field, since for each constant value assigned a particular flow field, since for each constant value assigned to to ΨΨa streamline can be a streamline can be drawn.drawn.

33


The actual numerical value associated with a particular streamliThe actual numerical value associated with a particular streamline is ne is not of particular significance, but the change in the value of not of particular significance, but the change in the value of ΨΨ is is related to the volume rate of flow.related to the volume rate of flow.For a unit depth, the flow rate across AB isFor a unit depth, the flow rate across AB is

For a unit depth, the flow rate across BC isFor a unit depth, the flow rate across BC is

12

y

y

y

y

2

1

2

1

2

1

ddyy

udyq ψ−ψ=ψ=∂ψ∂

== ∫∫∫ψ

ψ

12

x

x

x

x

1

2

2

1

2

1

ddxx

vdxq ψψψψ ψ

ψ−=−=

∂∂

−== ∫∫∫

34


Thus the volume flow rate between any two streamlines can be Thus the volume flow rate between any two streamlines can be written as the difference between the constant values of written as the difference between the constant values of ΨΨ defining defining two streamlines.two streamlines.The velocity will be relatively high wherever the streamlines arThe velocity will be relatively high wherever the streamlines are e close together, and relatively low wherever the streamlines are close together, and relatively low wherever the streamlines are far far apart.apart.

35

Stream Function Stream Function Cylindrical Coordinate SystemCylindrical Coordinate System

For a twoFor a two--dimensional, incompressible flow in the rdimensional, incompressible flow in the rθθplane, conservation of mass can be written as:plane, conservation of mass can be written as:

The velocity components can be related to the stream The velocity components can be related to the stream function, function, ΨΨ(r,(r,θθ) through the equation) through the equation

rvand

r1vr ∂

ψ∂−=

θ∂ψ∂

= θ

0vr

)rv( r =θ∂

∂+

∂∂ θ

36

Example 6.3 Stream FunctionExample 6.3 Stream Function

The velocity component in a steady, incompressible, two The velocity component in a steady, incompressible, two dimensional flow field aredimensional flow field are

Determine the corresponding stream function and show on a skDetermine the corresponding stream function and show on a sketch etch several streamlines. Indicate the direction of glow along the several streamlines. Indicate the direction of glow along the streamlines.streamlines.

4xv2yu ==

37


(y)fx2(x)fy 22

12 +−=ψ+=ψ

Cyx2 22 ++−=ψ

From the definition of the stream functionFrom the definition of the stream function

x4x

vy2y

u =∂ψ∂

−==∂ψ∂

=

For simplicity, we set C=0For simplicity, we set C=0

22 yx2 +−=ψΨΨ=0=0

ΨΨ≠≠0012/

xy 22

=ψ

−ψ

38

Conservation of Linear MomentumConservation of Linear Momentum

Applying NewtonApplying Newton’’s second law to control volumes second law to control volume

( )

amDt

VDm

zVw

yVv

xVu

tVm

tDmVDF

rr

rrrrrr

δ=δ=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

δ=δ

=δ

SYSDt

PDFr

r=

For a For a infinitesimal system of mass dminfinitesimal system of mass dm, what, what’’s the s the The The differential form of linear momentum equation?differential form of linear momentum equation?

VdVdmVP)system(V)system(Msystem ρ== ∫∫rrr

39

Forces Acting on Element Forces Acting on Element 1/21/2

The forces acting on a fluid element may be classified as body forces and surface forces; surface forces include normal forces and normal forces and tangentialtangential (shear) forces.

kFjFiF

kFjFiF

FFF

bzbybx

szsysx

BS

rrr

rrr

rrr

δ+δ+δ+

δ+δ+δ=

δ+δ=δSurface forces acting on a fluid Surface forces acting on a fluid element can be described in terms element can be described in terms of normal and shearing stresses.of normal and shearing stresses.

AFn

tn δδ

σδ 0lim→

=AF

t δδ

τδ

1

01 lim→

=AF

t δδ

τδ

2

02 lim→

=

40

Forces Acting on Element Forces Acting on Element 2/22/2

zyxgFzyxgFzyxgF

zyxzyx

F

zyxzyx

F

zyxzyx

F

zbz

yby

xbx

zzyzxzsz

zyyyxysy

zxyxxxsx

δδδρ=δ

δδδρ=δδδδρ=δ

δδδ⎟⎟⎠

⎞⎜⎜⎝

⎛∂σ∂

+∂τ∂

+∂τ∂

=δ

δδδ⎟⎟⎠

⎞⎜⎜⎝

⎛∂τ∂

+∂σ∂

+∂τ∂

=δ

δδδ⎟⎟⎠

⎞⎜⎜⎝

⎛∂τ∂

+∂τ∂

+∂σ∂

=δ

Equation of MotionEquation of Motion

41

Equation of MotionEquation of Motion

These are the differential equations of motion for anyThese are the differential equations of motion for anyfluid fluid satisfying the continuum assumptionsatisfying the continuum assumption..How to solve u,v,w ?How to solve u,v,w ?

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂σ∂

+∂τ∂

+∂τ∂

+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂τ∂

+∂σ∂

+∂τ∂

+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂τ∂

+∂τ∂

+∂σ∂

+ρ

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

zzyyxx maFmaFmaF δ=δδ=δδ=δ

42

Double Subscript Notation for StressesDouble Subscript Notation for Stresses

xyτ

The direction of the The direction of the normal to the plane normal to the plane on which the stress on which the stress actsacts

The direction of the stressThe direction of the stress

43

InviscidInviscid FlowFlow

Shearing stresses develop in a moving fluid because of the viscoShearing stresses develop in a moving fluid because of the viscosity sity of the fluid.of the fluid.For some common fluid, such as air and water, the viscosity is sFor some common fluid, such as air and water, the viscosity is small, mall, and therefore it seems reasonable to assume that under some and therefore it seems reasonable to assume that under some circumstances we may be able to simply neglect the effect of circumstances we may be able to simply neglect the effect of viscosity.viscosity.Flow fields in which the shearing stresses are assumed to be Flow fields in which the shearing stresses are assumed to be negligible are said to be negligible are said to be inviscidinviscid, , nonviscousnonviscous, or frictionless., or frictionless.

zzyyxxp σ=σ=σ=−Define the pressure, p, as the negative of the normal stressDefine the pressure, p, as the negative of the normal stress

44

EulerEuler’’s Equation of Motion s Equation of Motion

Under frictionless conditionfrictionless condition, the equations of motion are reduced to EulerEuler’’s Equation:s Equation:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂∂

−ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂∂

−ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂∂

−ρ

zww

ywv

xwu

tw

zpg

zvw

yvv

xvu

tv

ypg

zuw

yuv

xuu

tu

xpg

z

y

x

⎟⎟⎠

⎞⎜⎜⎝

⎛∇⋅+

∂∂

ρ=∇−ρ V)V(tVpg

rrr

r

45

Bernoulli Equation Bernoulli Equation 1/31/3

EulerEuler’’s equation for steady flows equation for steady flow along a streamline isalong a streamline is

V)V(pgrrr

∇⋅ρ=∇−ρ

( ) ( ) ( )VVVV21VV

rrrrrr×∇×−⋅∇=∇⋅

zgg ∇−=r

)()(2

VVVVpzgrrrr

×∇×−⋅∇=∇−∇− ρρρ

Selecting the coordinate system with the zSelecting the coordinate system with the z--axis vertical so that axis vertical so that the acceleration of gravity vector can be expressed asthe acceleration of gravity vector can be expressed as

Vector identity Vector identity ……..

46


( ) ( )[ ] sdVVsdzgsdV21sdp 2 rrrvrr

⋅×∇×=⋅∇+⋅∇+⋅ρ∇

( )VVrr

×∇× perpendicular to perpendicular to Vr( ) ( )VVzgV

21p 2

rr×∇×=∇+∇+

ρ∇

sdr⋅

With With kdzjdyidxsdrrsr

++=

dpdzzpdy

ypdx

xpsdp =

∂∂

+∂∂

+∂∂

=⋅∇r

47


( ) 0gdzVd21dp 2 =++

ρ

ttanconsgz2

Vdp 2

=++ρ∫

ttanconsgz2

Vp 2

=++ρ

( ) 0sdzgsdV21sdp 2 =⋅∇+⋅∇+⋅

ρ∇ rrr

Integrating Integrating ……

For steady For steady inviscidinviscid, incompressible fluid ( commonly called ideal , incompressible fluid ( commonly called ideal fluids) fluids) along a streamlinealong a streamline

Bernoulli equationBernoulli equation

48

IrrotationalIrrotational Flow Flow 1/41/4

Irrotation ? The irrotational condition is

In rectangular coordinates system

In cylindrical coordinates system

0V =×∇r

0xw

zu

zv

yw

yu

xv

=∂∂

−∂∂

=∂∂

−∂∂

=∂∂

−∂∂

0Vr1

rrV

r1

rV

zV

zVV

r1 rzrz =

θ∂∂

−∂∂

=∂∂

−∂∂

=∂∂

−θ∂

∂ θθ

49


A general flow field would not be A general flow field would not be irrotationalirrotational flow.flow.A special uniform flow field is an example of an A special uniform flow field is an example of an irrotationirrotationflowflow

50


A general flow fieldA solid body is placed in a uniform stream of fluid. Far away frA solid body is placed in a uniform stream of fluid. Far away from om the body remain uniform, and in this far region the flow is the body remain uniform, and in this far region the flow is irrotationalirrotational. . The flow around the body remains The flow around the body remains irrotationalirrotational except very near the except very near the boundary.boundary.Near the boundary the Near the boundary the velocity changes rapidly from velocity changes rapidly from zero at the boundary (nozero at the boundary (no--slip slip condition) to some relatively condition) to some relatively large value in a short distance large value in a short distance from the boundary.from the boundary. Chapter 9Chapter 9

51


A general flow fieldFlow from a large reservoir enters a pipe through a streamlined Flow from a large reservoir enters a pipe through a streamlined entrance where the velocity distribution is essentially uniform.entrance where the velocity distribution is essentially uniform. Thus, Thus, at entrance the flow is at entrance the flow is irrotationalirrotational. (b). (b)In the central core of the pipe the flow remains In the central core of the pipe the flow remains irrotationalirrotational for some for some distance.distance.The boundary layer will develop along the wall and grow in The boundary layer will develop along the wall and grow in thickness until it fills the pipe.thickness until it fills the pipe. Viscous forces are dominantViscous forces are dominant

Chapter 8Chapter 8

52

Bernoulli Equation for Bernoulli Equation for IrrotationalIrrotational Flow Flow 1/31/3

The Bernoulli equation forThe Bernoulli equation for steady, incompressible, and steady, incompressible, and inviscidinviscidflowflow isis

The equation can be applied betweenThe equation can be applied between any two points on the same any two points on the same streamlinestreamline. . In general,In general, the value of the constant will vary from the value of the constant will vary from streamline to streamlinestreamline to streamline..Under additionalUnder additional irrotationalirrotational conditioncondition, , the Bernoulli equation ?the Bernoulli equation ?Starting with EulerStarting with Euler’’s equation in vector forms equation in vector form

( ) ( )VVVV21kgp1V)V(

rrrrrrr×∇×−⋅∇=−∇

ρ−=∇⋅

ttanconsgz2

Vp 2

=++ρ

ZERO Regardless of the direction of ZERO Regardless of the direction of dsds

53


With irrotaionalirrotaional condition

( ) ( ) kgp1V21VV

21 2

rrr−∇

ρ−=∇=⋅∇

0V =×∇r

rdr⋅

( )

( ) ( ) 0gdzVd21dpgdzdpVd

21

rdkgrdp1rdV21

22

2

=++ρ

>>−ρ

−=>>

⋅−⋅∇ρ

−=⋅∇ vrvv

( ) ( )VVVV21kgp1V)V(

rrrrrrr×∇×−⋅∇=−∇

ρ−=∇⋅

54


Integrating for incompressible flowIntegrating for incompressible flow

This equation is valid between any two points in a steady, This equation is valid between any two points in a steady, incompressible, incompressible, inviscidinviscid, and , and irrotationalirrotational flow.flow.

ttanconsgz2

Vp 2

=++ρ

ttancongz2

Vdp 2

=++ρ∫

2

222

1

211 z

g2Vpz

g2Vp

++γ

=++γ

55

Velocity Potential Velocity Potential ΦΦ((x,y,z,t)x,y,z,t) 1/41/4

The stream function for twoThe stream function for two--dimensional incompressible dimensional incompressible flow isflow isΨΨ(x,y) which satisfies the continuous (x,y) which satisfies the continuous condtionscondtionsFor an For an irrotationalirrotational flow, the velocity components can be flow, the velocity components can be expressed in terms of a scalar function expressed in terms of a scalar function ψψ((x,y,z,tx,y,z,t)) asas

where where ψψ((x,y,z,tx,y,z,t)) is called the velocity potential.is called the velocity potential.

zw

yv

xu

∂φ∂

=∂φ∂

=∂φ∂

=

56


In vector formIn vector form

For an incompressible flow For an incompressible flow φ∇=V

r

LaplacianLaplacian operatoroperator

0V =⋅∇r

0zyx

VV 2

2

2

2

2

22 =

∂φ∂

+∂φ∂

+∂φ∂

=φ∇=⋅∇⇒φ∇=rr

For incompressible, For incompressible, irrotationalirrotational flowflow

Laplace’s equation

called a potential flowcalled a potential flow

57


InviscidInviscid, incompressible, , incompressible, irrotationalirrotational fields are governed fields are governed by Laplaceby Laplace’’s equation.s equation.This type flow is commonly called This type flow is commonly called a potential flowa potential flow..To complete the mathematical formulation of a given To complete the mathematical formulation of a given problem, boundary conditions have to be specified. These problem, boundary conditions have to be specified. These are usually velocities specified on the boundaries of the are usually velocities specified on the boundaries of the flow field of interest.flow field of interest.

58


In cylindrical coordinate, r, In cylindrical coordinate, r, θθ, and z, and z

zv

r1v

rv zr ∂

φ∂=

θ∂φ∂

=∂φ∂

= θ

0zr

1r

rrr

12

2

2

2

2 =∂φ∂

+θ∂φ∂

+⎟⎠⎞

⎜⎝⎛

∂φ∂

∂∂

59

Potential Flow TheoryPotential Flow Theory

Velocity potential ψψ ((x,y,z,tx,y,z,t)) exists only for irrotationalflow.Irrotationality may be a valid assumption for those regions of a flow in which viscous forces are negligible.In an irrotational flow, the velocity field may be defined by the potential function,ψψ((x,y,z,tx,y,z,t)), the theory is often referred to as potential flow theory.

60

LaplaceLaplace’’s Equation s Equation 1/21/2

For two-dimensional, incompressible flow

For two-dimensional, irrotational flow

)1(x

vy

u∂ψ∂

−=∂ψ∂

=

(1)(1) + + irrotationalirrotational condition condition ……

(2) + continuity equation (2) + continuity equation ……

0yx

0xv

yu

2

2

2

2

=∂ψ∂

+∂ψ∂

⇒=∂∂

−∂∂

)2(y

vx

u∂φ∂

=∂φ∂

=

0yx

0yv

xu

2

2

2

2

=∂φ∂

+∂φ∂

⇒=∂∂

+∂∂

61

LaplaceLaplace’’s Equation s Equation 2/22/2

For a two-dimensional incompressible flow, we can define a stream function Ψ; if the flow is also irrotational, Ψwill satisfy Laplace’s equation.For an irrotational flow, we can define a velocity potential Φ; if the flow is also incompressible, Φ will satisfy Laplace’s equation.Any function ψψ orΨ that satisfies Laplace’s equation represents a possible two-dimensional, incompressible, irrotational flow field.

62

ΦΦ and and ΨΨ 1/21/2

ForForΨΨ=constant=constant, , ddΨΨ =0=0 andand

The slope of a streamline The slope of a streamline –– a line of constanta line of constantΨΨ

Along a line of constantAlong a line of constantψψ, , ddψψ =0=0 andand

The slope of a potential line The slope of a potential line –– a line of constanta line of constantψψ

0dyy

dxx

d =∂ψ∂

+∂ψ∂

=ψ

uv

y/x/

dxdy

=∂ψ∂∂ψ∂

−=⎟⎠⎞

ψ

0dyy

dxx

d =∂φ∂

+∂φ∂

=φ

vu

y/x/

dxdy

−=∂φ∂∂φ∂

−=⎟⎠⎞

φ

Line of constant Ψ and constant ψψ are orthogonal.

63

ΦΦ and and ΨΨ 2/22/2

Flow net for 90° bend

The lines of constant The lines of constant ψψ (called (called equipotentialequipotential lines) are orthogonal to lines lines) are orthogonal to lines of constant of constant ΨΨ (streamlines) at all points (streamlines) at all points where they intersect..where they intersect..For any potential flow a For any potential flow a ““flow netflow net”” can be can be drawn that consists of a family of drawn that consists of a family of streamlines and streamlines and equipotentialequipotential lines. lines. Velocities can be estimated from the flow Velocities can be estimated from the flow net, since the velocity is inversely net, since the velocity is inversely proportional to the streamline spacingproportional to the streamline spacing

64

Example 6.4 Velocity Potential and Example 6.4 Velocity Potential and InviscidInviscid Flow Pressure Flow Pressure 1/21/2

The twoThe two--dimensional flow of a dimensional flow of a nonviscousnonviscous, incompressible fluid in , incompressible fluid in the vicinity of the 90the vicinity of the 90°° corner of Figure E6.4a is described by the corner of Figure E6.4a is described by the stream functionstream function

Where Where ΨΨ has units of mhas units of m22/s when r is in meters. (a) Determine, if /s when r is in meters. (a) Determine, if possible, the corresponding velocity potential. (b) If the presspossible, the corresponding velocity potential. (b) If the pressure at ure at point (1) on the wall is 30 point (1) on the wall is 30 kPakPa, what is the pressure at point (2)? , what is the pressure at point (2)? Assume the fluid density is 10Assume the fluid density is 1033 kg/mkg/m33 and the xand the x--y plane is y plane is horizontal horizontal –– that is, there is no difference in elevation between that is, there is no difference in elevation between points (1) and (2).points (1) and (2).

θ=ψ 2sinr2 2

65

Example 6.4 Velocity Potential and Example 6.4 Velocity Potential and InviscidInviscid Flow Pressure Flow Pressure 2/22/2

66

Example 6.4 Example 6.4 SolutionSolution1/21/2

θ−=∂ψ∂

−=

θ=θ∂ψ∂

=

θ 2sinr4r

v

2cosr4r1vr

)(f2cosr22cosr4r

v 12

r θ+θ=φ⇒θ=∂φ∂

=

)r(f2cosr22sinr4r1v 2

2 +θ=φ⇒θ−=θ∂φ∂

=θ

C2cosr2 2 +θ=φ

Let C=0

θ=φ 2cosr2 2

67


Bernoulli equation between points (1) and (2) with no elevation change

)VV(2

ppg2

Vpg2

Vp 22

2112

222

211 −

ρ+=⇒+

γ=+

γ

2222

2221

22r

2

s/m4...V

s/m16...V

vvV

==

==

+= θ

kPa36...p2 ==

xy4sincosr42cosr2 22 =θθ=θ=φ

68

Some Basic, Plane Potential FlowSome Basic, Plane Potential Flow

Since LaplaceSince Laplace’’s equation is linear, various solutions can be added to s equation is linear, various solutions can be added to obtain other solution obtain other solution –– that is , if that is , if ψψ11(x,y,z)(x,y,z) and and ψψ22(x,y,z)(x,y,z) are two are two solutions to Laplacesolutions to Laplace’’s equation, then s equation, then ψψ==ψψ11 + + ψψ22 is also solution.is also solution.The practical implication of this result is that if we have certThe practical implication of this result is that if we have certain ain basic solution we can combine them to obtain more complicated anbasic solution we can combine them to obtain more complicated and d interesting solutions.interesting solutions.Several basic velocity potentials, which describe some relativelSeveral basic velocity potentials, which describe some relatively y simple flows, will be determined.simple flows, will be determined.These basic velocity potential will be combined to represent These basic velocity potential will be combined to represent complicate flows.complicate flows.

69

Uniform Flow Uniform Flow 1/21/2

A uniform flow is a simplest plane flow for which the A uniform flow is a simplest plane flow for which the streamlines are all straight and parallel, and the magnitude streamlines are all straight and parallel, and the magnitude of the velocity is constant.of the velocity is constant.

CUx0y

,Ux

+=φ⇒=∂φ∂

=∂φ∂

u=U and v=0u=U and v=0

Uy0x

,Uy

=ψ⇒=∂ψ∂

=∂ψ∂

70

Uniform Flow Uniform Flow 2/22/2

For a uniform flow of constant velocity V, inclined to an For a uniform flow of constant velocity V, inclined to an angle angle αα to the xto the x--axis.axis.

x)cosV(y)sinV(x)sinV(y)cosV(

α−α−=φα−α=ψ

71

Source and Sink Source and Sink 1/21/2

For a For a sourcesource flow ( from origin flow ( from origin radiallyradially) with volume flow ) with volume flow rate per unit depth m rate per unit depth m (m=2(m=2ππ r r vvrr ))

rln2m

2m0v

r2mv r π

=φθπ

=ψ⇒=π

= θ

θ∂φ∂

=∂φ∂

=

∂ψ∂

−=θ∂ψ∂

=

θ

θ

r1vand

rv

rvand

r1v

r

r

72

Source and Sink Source and Sink 2/22/2

For a For a sinksink flow (toward origin flow (toward origin radiallyradially) with volume flow ) with volume flow rate per unit depth rate per unit depth mm

rln2m

2m0v

r2mvr π

−=φθπ

−=ψ⇒=π

−= θ

θ∂φ∂

=∂φ∂

=

∂ψ∂

−=θ∂ψ∂

=

θ

θ

r1vand

rv

rvand

r1v

r

r

73

Example 6.5 Potential Flow Example 6.5 Potential Flow -- SinkSink

A A nonviscousnonviscous, incompressible fluid flows between wedge, incompressible fluid flows between wedge--shaped shaped walls into a small opening as shown in Figure E6.5. The velocitywalls into a small opening as shown in Figure E6.5. The velocitypotential (in ftpotential (in ft22/s), which approximately describes this flow is/s), which approximately describes this flow is

Determine the volume rate Determine the volume rate of flow (per unit length) into of flow (per unit length) into the opening.the opening.

rln2−=φ

74


The components of velocity The components of velocity

0r1v

r2

rvr =

θ∂φ∂

=−=∂φ∂

= θ

s/ft05.13

...Rdvq 26/

0r −=

π−==θ= ∫

π

The The flowrateflowrate per unit widthper unit width

75

VortexVortex

A vortex represents a flow in which the streamlines are A vortex represents a flow in which the streamlines are concentric circles.concentric circles.Vortex motion can be either rotational or Vortex motion can be either rotational or irrotationalirrotational..For an For an irrotationalirrotational vortex (vortex (ccwccw, center at origin) with , center at origin) with vortex strength vortex strength KK

θ=φ−=ψ⇒

== θ

KrlnKrKv0v r

At r=0, the velocity At r=0, the velocity becomes infinite.becomes infinite. singularity

76

Free Vortex Free Vortex 1/21/2

Free (Free (IrrotationalIrrotational) vortex (a) is that rotation refers to the ) vortex (a) is that rotation refers to the orientation of a fluid element and not the path followed by orientation of a fluid element and not the path followed by the element.the element.A pair of small sticks were A pair of small sticks were placed in the flow field at placed in the flow field at location A, the sticks would location A, the sticks would rotate as they as they move to rotate as they as they move to location B. location B.

77

Free Vortex Free Vortex 2/22/2

One of the sticks, the one that is aligned the streamline, One of the sticks, the one that is aligned the streamline, would follow a circular path and rotate in a would follow a circular path and rotate in a counterclockwise directioncounterclockwise directionThe other rotates in a clockwise The other rotates in a clockwise direction due to the nature of the direction due to the nature of the flow field flow field –– that is, the part of the that is, the part of the stick nearest the origin moves stick nearest the origin moves faster than the opposite end.faster than the opposite end.The average velocity of the two The average velocity of the two sticks is zero.sticks is zero.

78

Forced VortexForced Vortex

If the flow were rotating as a If the flow were rotating as a rigid body, such that rigid body, such that vvθθ=K=K11r r where Kwhere K11 is a constant.is a constant.Force vortex is rotational and Force vortex is rotational and cannot be described with a cannot be described with a velocity potential.velocity potential.Force vortex is commonly Force vortex is commonly called a rotational vortex.called a rotational vortex.

79

Combined VortexCombined Vortex

A combine vortex is one with a forced vortex as a central A combine vortex is one with a forced vortex as a central core and a velocity distribution corresponding to that of a core and a velocity distribution corresponding to that of a free vortex outside the core.free vortex outside the core.

0

0

rrrv

rrrKv

≤ω=

>=

θ

θ

80

Circulation Circulation 1/31/3

Circulation Circulation ΓΓ is defined as theis defined as the line integral of the line integral of the tangential velocity component about any closed curvetangential velocity component about any closed curvefixed in the flow:fixed in the flow:

∫∫∫ ×∇=ω=⋅=ΓA

ZA

Zc

dA)V(dA2sdVrrr

sdrwhere the is an element vector tangent to the curve where the is an element vector tangent to the curve

and having length and having length dsds of the element of arc. Itof the element of arc. It’’s positive s positive corresponds to a c.c.w. direction of integration around the corresponds to a c.c.w. direction of integration around the curve.curve.

81


For For irrotationalirrotational flow , flow , ΓΓ =0=0

0dsddA)V(sdVccAc

=φ=⋅φ∇=×∇=⋅=Γ ∫∫∫∫rrrr

For For irrotationalirrotational flow , flow , ΓΓ =0=0

The circulation around any path that does not include the singular point at the origin will be zero.

The circulation around any path that does not include the singular point at the origin will be zero.

82


For free vortexFor free vortex

The circulation around any path that encloses singularities will be nozero.The circulation around any path that encloses singularities will be nozero.

πΓ=

π=θ=Γ ∫π

/2K

K2)rd(rK2

0

θπΓ

=φπΓ

−=ψ2

rln2

rKv =θ

83

Example 6.6 Potential Flow Example 6.6 Potential Flow –– Free VortexFree Vortex

A liquid drains from a large tank through a small opening as A liquid drains from a large tank through a small opening as illustrated in Figure E6.6. A vortex forms whose velocity illustrated in Figure E6.6. A vortex forms whose velocity distribution away from the tank opening can be approximated as tdistribution away from the tank opening can be approximated as that hat of a free vortex having a velocity potentialof a free vortex having a velocity potential

θπΓ

=φ2

Determine an expression Determine an expression relating the surface shape relating the surface shape to the strength of the to the strength of the vortex as specified by the vortex as specified by the circulation circulation ΓΓ..

84


0pp 21 ==

Since the free vortex represents an Since the free vortex represents an irrotationalirrotational flow field, the flow field, the Bernoulli equationBernoulli equation

2

222

1

211 z

g2Vpz

g2Vp

++γ

=++γ

At free surface

s

22

21 z

g2V

g2V

+=

r2r1v

πΓ

=θ∂φ∂

=θ

Far from the origin at point (1), V1=vθ=0

gr2z 22

2

s πΓ

−=

85

Doublet Doublet 1/21/2

For a doublet ( produced mathematically by allowing a For a doublet ( produced mathematically by allowing a source and a sink of numerically equal strength to merge) source and a sink of numerically equal strength to merge) with a strength mwith a strength m

( )212m

θ−θπ

−=ψ

The combined stream function for the pair isThe combined stream function for the pair is

( )21

2121 tantan1

tantantanm

2tanθθ+θ−θ

=θ−θ=⎟⎠⎞

⎜⎝⎛ πψ−

( ) ( )

( ) ⎟⎠⎞

⎜⎝⎛

−θ

π−=ψ⇒

−θ

=⎟⎠⎞

⎜⎝⎛ πψ−⇒

+θθ

=θ−θθ

=θ

−22

122

21

arsinar2tan

2m

arsinar2

m2tan

acosrsinrtanand

acosrsinrtan

86

Doublet Doublet 2/22/2

( )2222 arsinmar

arsinar2

2m

−πθ

−=−

θπ

−=ψaa→→0 0

The soThe so--called doublet is formed by letting acalled doublet is formed by letting a→→00，，mm→∞→∞

r1

arr

22 →−

rsinK θ

−=ψ

π=

maK

rcosK θ

=φ

87

Streamlines for a DoubletStreamlines for a Doublet

Plots of lines of constant Plots of lines of constant ΨΨ reveal that the streamlines for reveal that the streamlines for a doublet are circles through the origin tangent to the x a doublet are circles through the origin tangent to the x axis.axis.

88

SummarySummary

89

Superposition of Elementary Plane Superposition of Elementary Plane Flows Flows 1/21/2

Potential flows are governed by Laplace’s equation, which is a linear partial differential equation.Various basic velocity potentials and stream function, ψψand Ψ, can be combined to form new potentials and stream functions.

213213 φ+φ=φψ+ψ=ψ

90

Superposition of Elementary Plane Superposition of Elementary Plane Flows Flows 2/22/2

Any streamline in an inviscid flow field can be considered as a solid boundary, since the conditions along a solid boundary and as streamline are the same – that is, there is no flow through the boundary or the streamline.We can combine some of the basic velocity potentials or stream functions to yield a streamline that corresponds to a particular body shape of interest, that combination can be used to describe in detail the flow around that body.

Methods of superpositionMethods of superposition

91

HalfHalf--Body : Uniform Stream + Source Body : Uniform Stream + Source 1/41/4

rln2mcosUr

2msinUr

sourceflowuniform

sourceflowuniform

π+=φ+φ=φ

θπ

+θ=ψ+ψ=ψ

−

−

θ−=∂ψ∂

−=

π+θ=

θ∂ψ∂

=

θ sinUr

v

r2mcosU

r1vr

The stagnation point occurs at x=The stagnation point occurs at x=--bb

U2mb0

b2mUvr π

=⇒=π−

+=The combination of a uniform flow and a The combination of a uniform flow and a source can be used to describe the flow source can be used to describe the flow around a streamlined body placed in a around a streamlined body placed in a uniform stream.uniform stream.

92


2m

stagnation =ψ

θθ−π

=

θ+θ=π

sin)(br

bUsinUrbU

bU2m

π=

The value of the stream function at the stagnation point can be The value of the stream function at the stagnation point can be obtained by evaluating obtained by evaluating ΨΨ at r=b at r=b θθ==ππ

The equation of the streamline passing through the stagnation The equation of the streamline passing through the stagnation point ispoint is

The streamline can be replaced by a solid boundary. The streamline can be replaced by a solid boundary. The body is open at the downstream end, and thus is The body is open at the downstream end, and thus is called a HALFcalled a HALF--BODY. The combination of a uniform BODY. The combination of a uniform flow and a source can be used to describe the flow flow and a source can be used to describe the flow around a streamlined body placed in a uniform stream.around a streamlined body placed in a uniform stream.

93


)(bysin

)(br θ−π=⇒θθ−π

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+θ+=⎟

⎠⎞

⎜⎝⎛π

+π

θ+=+= θ 2

22

2222

r2

rbcos

rb21U

r2m

rcosUmUvvV

As As θθ 0 or 0 or θθ= = ππ the halfthe half--width approaches width approaches ±±bbππ.. The width The width of the halfof the half--body asymptotically approach 2body asymptotically approach 2ππb.b.The velocity components at any pointThe velocity components at any point

θ−=∂ψ∂

−=π

+θ=θ∂ψ∂

= θ sinUr

vr2

mcosUr1vr

U2mbπ

=

94


With the velocity known, the pressure at any point can be determined from the Bernoulli equation

Where elevation change have been neglected.

220 V

21pU

21p ρ+=ρ+

Far from the bodyFar from the body

95

Example 6.7 Potential Flow Example 6.7 Potential Flow –– HalfHalf--BodyBody

The shape of a hill arising from a plain can be approximated witThe shape of a hill arising from a plain can be approximated with h the top section of a halfthe top section of a half--body as is illustrated in Figure E6.7a. The body as is illustrated in Figure E6.7a. The height of the hill approaches 200 ft as shown. (a) When a 40 mi/height of the hill approaches 200 ft as shown. (a) When a 40 mi/hr hr wind blows toward the hill, what is the magnitude of the air velwind blows toward the hill, what is the magnitude of the air velocity ocity at a point on the hill directly above the origin [point (2)]? (bat a point on the hill directly above the origin [point (2)]? (b) What ) What is the elevation of point (2) above the plain and what is the is the elevation of point (2) above the plain and what is the difference in pressure between point (1) on the plain far from tdifference in pressure between point (1) on the plain far from the hill he hill and point (2)? Assume an air density of 0.00238 slugs/ftand point (2)? Assume an air density of 0.00238 slugs/ft33??

96


⎟⎟⎠

⎞⎜⎜⎝

⎛+θ+= 2

222

rbcos

rb21UV

The velocity isThe velocity is

At point (2), At point (2), θθ==ππ/2/2

2b

sin)(br π=

θθ−π

=

hr/mi4.47...41U)2/b(

b1UV 22

2

222 ==⎟

⎠

⎞⎜⎝

⎛

π+=⎟

⎟⎠

⎞⎜⎜⎝

⎛

π+=

The elevation at (2) above the plain isThe elevation at (2) above the plain is ft1002

ft2002by2 ==

π=

97


From the Bernoulli equationFrom the Bernoulli equation

psi0647.0ft/lb31.9...)yy()VV(2

pp

zg2

Vpzg2

Vp

212

21

2221

2

222

1

211

===−γ+−ρ

=−

++γ

=++γ

s/ft5.69hr/s3600mi/ft5280)hr/mi4.47(V

s/ft7.58hr/s3600mi/ft5280)hr/mi40(V

2

1

=⎟⎠⎞

⎜⎝⎛=

=⎟⎠⎞

⎜⎝⎛=

98

RankineRankine Oval: Uniform Stream + Doublet Oval: Uniform Stream + Doublet 1/31/3

( )

( )21

21

rlnrln2mcosUr

2msinUr

−π

−θ=φ

θ−θπ

−θ=ψ

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+π−=ψ

⎟⎠

⎞⎜⎝

⎛

−

θπ

−θ=ψ

−

−

2221

221

ayxay2tan

2mUy

arsinar2tan

2msinUr

The corresponding streamlines for this flow field are obtained bThe corresponding streamlines for this flow field are obtained by y setting setting ΨΨ=constant. It is discovered that the streamline forms a =constant. It is discovered that the streamline forms a closed body of length 2closed body of length 2ll and width 2h.and width 2h. RankineRankine ovalsovals

99


2/12/12 1

Uam

aa

Uma

⎟⎠⎞

⎜⎝⎛ +π

=⇒⎟⎠⎞

⎜⎝⎛ +π

=l

l

The stagnation points occur at the upstream and downstream ends The stagnation points occur at the upstream and downstream ends of of the body. These points can be located by determining where alongthe body. These points can be located by determining where alongthe x axis the velocity is zero.the x axis the velocity is zero.The stagnation points correspond to the points where the uniformThe stagnation points correspond to the points where the uniformvelocity, the source velocity, and the sink velocity all combinevelocity, the source velocity, and the sink velocity all combine to to give a zero velocity. give a zero velocity. The locations of the stagnation points depend on the value of a,The locations of the stagnation points depend on the value of a, m, m, and U.and U.

The body halfThe body half--length 2length 2ll DimensionlessDimensionless

100


⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ π

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛=⇒

π−=

ah

mUa2tan1

ah

21

ah

mUh2tan

a2ahh

222

The body halfThe body half--width, h, can be obtained by determining the value of width, h, can be obtained by determining the value of y where the y axis intersects the y where the y axis intersects the ΨΨ=0 streamline. Thus, with =0 streamline. Thus, with ΨΨ=0, =0, x=0, and y=h.x=0, and y=h.

The body halfThe body half--width 2hwidth 2h

DimensionlessDimensionless

101

Flow around a Circular Cylinder Flow around a Circular Cylinder 1/41/4

rcosKcosUr

rsinKsinUr

θ+θ=φ

θ−θ=ψ

When the distance between the sourceWhen the distance between the source--sink pair approaches zero, the sink pair approaches zero, the shape of the shape of the rankinerankine oval becomes more blunt and in fact oval becomes more blunt and in fact approaches a circular shape. approaches a circular shape.

θ⎟⎟⎠

⎞⎜⎜⎝

⎛+=φ

θ⎟⎟⎠

⎞⎜⎜⎝

⎛−=ψ

cosra1Ur

sinra1Ur

2

2

2

2

Ψ=constant for r=aΨ=0 for r=a K=Ua2

In order for the stream function to represent flow around a circular cylinder

102


The velocity componentsThe velocity components

θ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

∂ψ∂

−=θ∂φ∂

=

θ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

θ∂ψ∂

=∂φ∂

=

θ sinra1U

rr1v

cosra1U

r1

rv

2

2

2

2

r

On the surface of the cylinder (r=a)

θ−==

θ sinU2v0vr

The pressure distribution on the cylinder surface can be obtaineThe pressure distribution on the cylinder surface can be obtained d from the Bernoulli equationfrom the Bernoulli equation

22ss

20 v

21pU

21p θρ+=ρ+

Far from the bodyFar from the body

)sin41(U21pp 22

0s θ−ρ+=

θ−=θ sinU2v

103


On the upstream part of the On the upstream part of the cylinder, there is approximate cylinder, there is approximate agreement between the potential agreement between the potential flow and the experimental results. flow and the experimental results. Because of the viscous boundary Because of the viscous boundary layer that develops on the cylinder, layer that develops on the cylinder, the main flow separates from the the main flow separates from the surface of the cylinder, leading to surface of the cylinder, leading to the large difference between the the large difference between the theoretical, frictionless solution theoretical, frictionless solution and the experimental results on the and the experimental results on the downstream side of the cylinder.downstream side of the cylinder.

104


∫∫

π

π

θθ−=

θθ−=

2

0sy

2

0sx

adsinpF

adcospF

105

Flow around a Circular Cylinder + Free Flow around a Circular Cylinder + Free Vortex Vortex 1/41/4

θπΓ

+θ⎟⎟⎠

⎞⎜⎜⎝

⎛+=φ

πΓ

−θ⎟⎟⎠

⎞⎜⎜⎝

⎛−=ψ

2cos

ra1Urrln

2sin

ra1Ur 2

2

2

2

Adding a free vortex to the stream function or velocity Adding a free vortex to the stream function or velocity potential for the flow around a cylinder. potential for the flow around a cylinder.

The circle rThe circle r==a will still be a streamline, since the a will still be a streamline, since the streamlines for the added free vortex are all circular.streamlines for the added free vortex are all circular.

a2sinU2

rv

ar πΓ

+θ−=∂ψ∂

−==

θThe tangential velocity The tangential velocity on the surface of the on the surface of the cylindercylinder

106


This type of flow could be approximately created by This type of flow could be approximately created by placing a rotating cylinder in a uniform stream. placing a rotating cylinder in a uniform stream. Because of the presence of viscosity in any real fluid, the Because of the presence of viscosity in any real fluid, the fluid in contacting with the rotating cylinder would rotate fluid in contacting with the rotating cylinder would rotate with the same velocity as the cylinder, and the resulting with the same velocity as the cylinder, and the resulting flow field would resemble that developed by the flow field would resemble that developed by the combination of a uniform flow past a cylinder and a free combination of a uniform flow past a cylinder and a free vortex.vortex.

107


A variety of streamline patterns A variety of streamline patterns can be developed, depending can be developed, depending on the vortex strength on the vortex strength ГГ..The location of stagnation The location of stagnation points on a circular cylinder (a) points on a circular cylinder (a) without circulation; (b, c, d) without circulation; (b, c, d) with circulation.with circulation.

Ua4sin stag π

Γ=θ

stag0v θ=θ=θ

108


For the cylinder with circulation, the surface pressure, For the cylinder with circulation, the surface pressure, ppss, is , is obtained from the Bernoulli equationobtained from the Bernoulli equation

⎟⎟⎠

⎞⎜⎜⎝

⎛πΓ

−π

θΓ+θ−ρ+=

⎟⎠⎞

⎜⎝⎛

πΓ

+θ−ρ+=ρ+

222

222

0s

2

s2

0

Ua4aUsin2sin41U

21pp

a2sinU2

21pU

21p

UTadsinpF

0adcospF2

0 sy

2

0 sx

ρ−=θθ−=

=θθ−=

∫∫

π

πDragDrag

LiftLift

109

Example 6.8 Potential Flow Example 6.8 Potential Flow –– Cylinder Cylinder 1/21/2

When a circular cylinder is placed in a uniform stream, a stagnaWhen a circular cylinder is placed in a uniform stream, a stagnation tion point is created on the cylinder as is shown in Figure E6.8a. Ifpoint is created on the cylinder as is shown in Figure E6.8a. If a a small hole is located at this point, the stagnation pressure, small hole is located at this point, the stagnation pressure, ppstagstag, can , can be measured and used to determine the approach velocity, U. (a) be measured and used to determine the approach velocity, U. (a) Show how Show how ppstagstag and U are related. (b) If the cylinder is misaligned and U are related. (b) If the cylinder is misaligned by an angle by an angle αα (Figure E6.8b), but the measured pressure still (Figure E6.8b), but the measured pressure still interpreted as the stagnation pressure, determine an expression interpreted as the stagnation pressure, determine an expression for for the ratio of the true velocity, U, to the predicted velocity, Uthe ratio of the true velocity, U, to the predicted velocity, U’’. Plot . Plot this ratio as a function of this ratio as a function of αα for the range for the range --20 20 °≦α≦°≦α≦2020°°..

110

Example 6.8 Potential Flow Example 6.8 Potential Flow –– Cylinder Cylinder 2/22/2

111


The Bernoulli equation between a point on the stagnation streamlThe Bernoulli equation between a point on the stagnation streamline ine upstream from the cylinder and the stagnation pointupstream from the cylinder and the stagnation point

2/1

0stag

stag2

0

)pp(2U

pg2

Up

⎥⎦

⎤⎢⎣

⎡−

ρ=⇒

γ=+

γ

If the cylinder is misaligned by an angle, If the cylinder is misaligned by an angle, αα, the pressure actually , the pressure actually measured, pmeasured, paa, will be different from the stagnation pressure., will be different from the stagnation pressure.

The difference between the pressure at the The difference between the pressure at the stagnation point and the upstream pressurestagnation point and the upstream pressure

1/2

0a

0stag2/1

0a pppp

)(predictedU'U(true))pp(2'U ⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=⇒⎥⎦

⎤⎢⎣

⎡−

ρ=

112


On the surface of the cylinder (r=a)On the surface of the cylinder (r=a) θ−=θ sinU2v

2a

20 )sinU2(

21pU

21p α−ρ+=ρ+

2/12

20stag

220a

)sin41('U

U

U21pp

)sin41(U21pp

α−=

ρ=−

α−ρ=−

The Bernoulli equation between a point upstream if the cylinder The Bernoulli equation between a point upstream if the cylinder and and the point on the cylinder where r=a, the point on the cylinder where r=a, θθ==αα. .

113

Viscous FlowViscous Flow

To incorporate viscous effects into the differential analysis of fluid motion

StressStress--Deformation Relationship Deformation Relationship

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂σ∂

+∂

τ∂+

∂τ∂

+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂

τ∂+

∂

σ∂+

∂

τ∂+ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ=∂τ∂

+∂

τ∂+

∂σ∂

+ρ

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

General equation of motionGeneral equation of motion

114

StressStress--Deformation Relationship Deformation Relationship 1/21/2

The stresses must be The stresses must be expressed in terms of the expressed in terms of the velocity and pressure velocity and pressure field.field.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ=τ=τ

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

µ=τ=τ

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ=τ=τ

∂∂

µ+⋅∇µ−−=σ

∂∂

µ+⋅∇µ−−=σ

∂∂

µ+⋅∇µ−−=σ

zv

yw

zu

xw

yu

xv

zw2V

32p

yv2V

32p

xu2V

32p

zyyz

zxxz

yxxy

zz

yy

xx

r

r

r

Cartesian coordinates

115

StressStress--Deformation Relationship Deformation Relationship 2/22/2

⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

µ=τ=τ

⎟⎠⎞

⎜⎝⎛

θ∂∂

+∂∂

µ=τ=τ

⎟⎟⎠

⎞⎜⎜⎝

⎛θ∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

µ=τ=τ

∂∂

µ+−=σ

⎟⎠⎞

⎜⎝⎛ +

θ∂∂

µ+−=σ

∂∂

µ+−=σ

θθθ

θθθ

θθθ

rv

zv

vr1

zv

vr1

rv

rr

zv2p

rvv

r12p

rv2p

zrzrrz

zzz

rrr

zzz

r

rrr

Introduced into the differentialIntroduced into the differentialequation of motionequation of motion……..

Cylindrical polar coordinates

116

The The NavierNavier--Stokes Equations Stokes Equations 1/51/5

These obtained equations of motion are called the These obtained equations of motion are called the NavierNavier--Stokes Equations.Stokes Equations.

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

µ∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

µ∂∂

+∂∂

−ρ=ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅∇−

∂∂

µ∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ∂∂

+∂∂

−ρ=ρ

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

µ∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

µ∂∂

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ⋅∇−

∂∂

µ∂∂

+∂∂

−ρ=ρ

V32

zw2

zyw

zv

yzxu

xw

xzpg

DtDw

yw

zv

zV

32

yv2

yxv

yu

xypg

DtDv

zu

xw

zxv

yu

yV

32

xu2

xxpg

DtDu

z

y

x

r

r

r

Cartesian coordinatesCartesian coordinates

117⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

θ∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

µ+ρ+∂∂

−=

⎟⎠⎞

⎜⎝⎛

∂∂

+θ∂

∂++

∂∂

+∂∂

ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

θ∂∂

+θ∂

∂+−⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

µ+ρ+θ∂∂

−=

⎟⎠⎞

⎜⎝⎛

∂∂

++θ∂

∂+

∂∂

+∂∂

ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂+

θ∂∂

−θ∂

∂+−⎟

⎠⎞

⎜⎝⎛

∂∂

∂∂

µ+ρ+∂∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+−θ∂

∂+

∂∂

+∂∂

ρ

θ

θθθθθ

θθθθθθ

θ

θθ

2z

2

2z

2

2z

z

zz

zzr

z

2

2r

22

2

22

zr

r

2r

2

22r

2

22rr

r

rz

2rr

rr

zvv

r1

rvr

rr1g

zP

zvvv

rv

rvv

tv

zvv

r2v

r1

rv

rvr

rr1gp

r1

zvv

rvvv

rv

rvv

tv

zvv

r2v

r1

rv

rvr

rr1g

rp

zvv

rvv

rv

rvv

tv


Cylindrical polar coordinatesCylindrical polar coordinates

118


UnderUnder incompressible flow with constant viscosity incompressible flow with constant viscosity conditionsconditions, , the the NavierNavier--Stokes equations are reduced to:Stokes equations are reduced to:

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂µ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂µ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂

∂+

∂

∂µ+ρ+

∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

2

2

2

2

2

2

z

2

2

2

2

2

2

y

2

2

2

2

2

2

x

zw

yw

xwg

zp

zww

ywv

xwu

tw

zv

yv

xvg

yp

zvw

yvv

xvu

tv

zu

yu

xug

xp

zuw

yuv

xuu

tu

119


UndeUnder r frictionless conditionfrictionless condition, , the equations of motion are the equations of motion are reduced toreduced to EulerEuler’’s Equations Equation::

z

y

x

gzp

zww

ywv

xwu

tw

gyp

zvw

yvv

xvu

tv

gxp

zuw

yuv

xuu

tu

ρ+∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

ρ+∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

ρ+∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

ρ

pgDt

VD∇−ρ=ρ

rr

120


The The NavierNavier--Stokes equations apply to both laminar and Stokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity turbulent flow, but for turbulent flow each velocity component fluctuates randomly with respect to time and component fluctuates randomly with respect to time and this added complication makes an analytical solution this added complication makes an analytical solution intractable.intractable.The exact solutions referred to are for laminar flows in The exact solutions referred to are for laminar flows in which the velocity is either independent of time (steady which the velocity is either independent of time (steady flow) or dependent on time (unsteady flow) in a wellflow) or dependent on time (unsteady flow) in a well--defined manner.defined manner.

121

Some Simple Solutions for Viscous, Some Simple Solutions for Viscous, Incompressible FluidsIncompressible Fluids

A principal difficulty in solving the A principal difficulty in solving the NavierNavier--Stokes Stokes equations is because of their nonlinearity arising from the equations is because of their nonlinearity arising from the convective acceleration termsconvective acceleration terms..There are no general analytical schemes for solving There are no general analytical schemes for solving nonlinear partial differential equations.nonlinear partial differential equations.There are a few special cases for which the convective There are a few special cases for which the convective acceleration vanishes. In these cases exact solution are acceleration vanishes. In these cases exact solution are often possible.often possible.

122

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 1/61/6

Consider flow between the two horizontal, infinite Consider flow between the two horizontal, infinite parallel plate.parallel plate.

For this geometry the fluid particle move in the x For this geometry the fluid particle move in the x direction parallel to the pates, and there is no velocity direction parallel to the pates, and there is no velocity in the y or z direction in the y or z direction –– that is, v=0 and w=0.that is, v=0 and w=0.

123


From the continuity equation that From the continuity equation that ∂∂u/u/∂∂xx=0.=0.There would be no variation of u in the z direction for There would be no variation of u in the z direction for infinite plates, and for steady flow so that u=infinite plates, and for steady flow so that u=u(yu(y).).

The The NavierNavier--Stokes equations reduce toStokes equations reduce to

zp0g

yp0

yu

xp0 2

2

∂∂

−=ρ−∂∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

µ+∂∂

−=

124


⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂µ+

∂∂

−=

∂∂

−=

ρ−∂∂

−=

2

2

yu

xp0

zp0

gyp0

( )xfgyp 1+ρ−=IntegratingIntegrating

?c?ccycyxp

21u 2121

2 ++⎟⎠⎞

⎜⎝⎛∂∂

µ=

IntegratingIntegrating

125


With the boundary conditions u=0 at y=With the boundary conditions u=0 at y=--h u=0 at y=hh u=0 at y=h

212 h

xp

21c,0c ⎟

⎠⎞

⎜⎝⎛∂∂

µ−==

( )22 hyxp

21u −⎟

⎠⎞

⎜⎝⎛∂∂

µ=Velocity distributionVelocity distribution

126


Shear stress distributionShear stress distribution

Volume flow rate Volume flow rate

yxp

yx ⎟⎠⎞

⎜⎝⎛∂∂

=τ

⎟⎠⎞

⎜⎝⎛∂∂

µ−=−⎟

⎠⎞

⎜⎝⎛∂∂

µ== ∫∫ −− x

p3h2dy)hy(

xp

21udyq

3h

h

22h

h

l

ll

µ∆

=>>

∆−=

−==

∂∂

3ph2q

pppttanconsxp

3

12

127


Average velocityAverage velocity

Point of maximum velocityPoint of maximum velocitylµ∆

==3

phh2

qV2

average

0dydu

= at y=0

average

2

max V23

xp

2hUuu =⎟

⎠⎞

⎜⎝⎛∂∂

µ−===

128

CouetteCouette Flow Flow 1/31/3

Since only theSince only the boundary conditions have changedboundary conditions have changed, , there there isis no need to repeat the entire analysisno need to repeat the entire analysis of the of the ““both both plates stationaryplates stationary”” case.case.

?c?ccycyxp

21u 2121

2 ++⎟⎠⎞

⎜⎝⎛∂∂

µ=

129


The boundary conditions for the moving plate case areThe boundary conditions for the moving plate case areu=0 at y=0u=0 at y=0u=U at y=bu=U at y=b

0cbxp

21

bUc 21 =⎟

⎠⎞

⎜⎝⎛∂∂

µ−=⇒

Velocity distributionVelocity distribution

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛∂∂

µ−=

⎟⎠⎞

⎜⎝⎛∂∂

µ−⎟

⎠⎞

⎜⎝⎛∂∂

µ+=

yb1

by

xp

U2b

by

Uu

byxp

21y

xp

21

bUyu

2

2

⎟⎠⎞

⎜⎝⎛∂∂

µ=

xp

U2bP

2

130


Simplest type of Couette flow

byUu0

xp

=⇒=∂∂

This flow can be approximated by the flow between closely spaced concentric cylinder is fixed and the other cylinder rotates with a constant angular velocity.

Flow in the narrow gap of a journal bearing.

)rr/(rrrbrU

oi

ioi−ωµ=τ

−=ω=

131

Example 6.9 Plane Example 6.9 Plane CouetteCouette FlowFlow

A wide moving belt passes through a A wide moving belt passes through a container of a viscous liquid. The belt container of a viscous liquid. The belt moves vertically upward with a constant moves vertically upward with a constant velocity, Vvelocity, V00, as illustrated in Figure , as illustrated in Figure E6.9a. Because of viscous forces the belt E6.9a. Because of viscous forces the belt picks up a film of fluid of thickness h. picks up a film of fluid of thickness h. Gravity tends to make the fluid drain Gravity tends to make the fluid drain down the belt. Use the down the belt. Use the NavierNavier Stokes Stokes equations to determine an expression for equations to determine an expression for the average velocity of the fluid film as it the average velocity of the fluid film as it is dragged up the belt. Assume that the is dragged up the belt. Assume that the flow is laminar, steady, and fully flow is laminar, steady, and fully developed.developed.

132


0yv=

∂∂

0zp0

xp

=∂∂

=∂∂

Since the flow is assumed to be fully developed, the only velociSince the flow is assumed to be fully developed, the only velocity ty component is in the y direction so that u=w=0.component is in the y direction so that u=w=0.From the continuity equationFrom the continuity equation

, and for steady flow, so that v=v=v(xv(x))

2

2

dxvdg0 µ+ρ−= IntegratingIntegrating

⎟⎠⎞

⎜⎝⎛µ=τ

+µγ

=

dxdv

cxdxdv

xy

1

133


µγ

−=hc1

IntegratingIntegratinghxat0xy ==τ

22 cx

2hx

2v +

µγ

−µγ

=

0xatVv 0 ==

02 Vx

2hx

2v +

µγ

−µγ

=

134

Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 1/51/5

Consider the flow through a horizontal circular tube of Consider the flow through a horizontal circular tube of radius R.radius R.

0z

v0v,0v zr =

∂∂

⇒== θ ( )rvv zz =

135

Steady, Laminar Flow in Circular TubesSteady, Laminar Flow in Circular Tubes 2/52/5

NavierNavier –– Stokes equation reduced to Stokes equation reduced to

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

µ+∂∂

−=

θ∂∂

−θρ−=

∂∂

−θρ−=

rvr

rr1

zp0

pr1sing0

rpsing0

z

( ) ( )zfsinrgp 1+θρ−=

( )zfgyp 1+ρ−=IntegratingIntegrating

IntegratingIntegrating

?c?ccrlncrzp

41v 2121

2z ++⎟

⎠⎞

⎜⎝⎛∂∂

µ=

136

Steady, Laminar Flow in Circular TubesSteady, Laminar Flow in Circular Tubes 3/53/5

At r=0, the velocity At r=0, the velocity vvzz is finite. At r=R, the velocity is finite. At r=R, the velocity vvzzis zero.is zero.

221 R

zp

41c,0c ⎟

⎠⎞

⎜⎝⎛∂∂

µ−==

( )22z Rr

zp

41v −⎟

⎠⎞

⎜⎝⎛∂∂

µ=

Velocity distributionVelocity distribution

137


The shear stress distributionThe shear stress distribution

Volume flow rateVolume flow rate

⎟⎠⎞

⎜⎝⎛∂∂

=µ=τxp

2r

drdu

rx

⎟⎠⎞

⎜⎝⎛∂∂

µπ

−==π= ∫ zp

8R.....rdr2uQ

4R

0z

ll

ll

µ∆π

=⎟⎠⎞

⎜⎝⎛ ∆

µπ

=⎟⎠⎞

⎜⎝⎛∂∂

µπ

−=>>

∆−=−

==∂∂

128Dpp

8R

zp

8RQ

/pppttanconszp

444

12

138


Average velocityAverage velocity

Point of maximum velocityPoint of maximum velocity

lµ∆

=π

==8

pRRQ

AQV

2

2average

0dr

dv z = at r=0at r=0

2

max

zaverage

2

max Rr1

vvV2

4pRv ⎟

⎠⎞

⎜⎝⎛−=⇒=

µ∆

−=l

139

Steady, Axial, Laminar Flow in an Annulus Steady, Axial, Laminar Flow in an Annulus 1/21/2

Boundary conditionsBoundary conditionsvvzz = 0 , at r = = 0 , at r = rroovvzz = 0 , at r = = 0 , at r = rrii

For steady, laminar flow in circular tubesFor steady, laminar flow in circular tubes

?c?ccrlncrzp

41v 2121

2z ++⎟

⎠⎞

⎜⎝⎛∂∂

µ=

140

Steady, Axial, Laminar Flow in an Annulus Steady, Axial, Laminar Flow in an Annulus 2/22/2

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −+−⎟

⎠⎞

⎜⎝⎛∂∂

µ=

oio

2o

2i2

o2

z rrln

)r/rln(rrrr

zp

41v

The volume rate of flowThe volume rate of flow

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−−⎟

⎠⎞

⎜⎝⎛∂∂

µπ

−=π= ∫ )r/rln()rr(rr

zp

8dr)r2(vQ

io

22i

2o4

i4

or

rz

o

i

The maximum velocity occurs at r=The maximum velocity occurs at r=rrmm

2/1

io

2i

2o

m )r/rln(2rrr

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=0

rvz =∂∂

The velocity distributionThe velocity distribution

fluid 06

Documents

xx yx zx

satisfy laplaces

general flow

ro ri

basic velocity

irrotational

volume flow

dimensional