FUNDAMENTALS OF FLUID MECHANICSFLUID ...cau.ac.kr/~jjang14/FME/Chap6.pdfChapter 6 Flow AnalysisChapter 6 Flow Analysis Using Differential MethodsDifferential Methods 1 MAIN TOPICSMAIN

FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICSFLUID MECHANICSFLUID MECHANICS

Chapter 6 Flow AnalysisChapter 6 Flow AnalysisChapter 6 Flow Analysis Chapter 6 Flow Analysis UsingUsing Differential MethodsDifferential MethodsUsing Using Differential MethodsDifferential Methods

1

MAIN TOPICSMAIN TOPICSMAIN TOPICSMAIN TOPICS

II Fl id ElFl id El M iM iI.I. Fluid Element Fluid Element MotionMotionII.II. Conservation of MassConservation of Mass (Continuity equation) and C(Continuity equation) and Conservation of onservation of

Linear MomentumLinear Momentum (Navier(Navier Stokes Equation)Stokes Equation)Linear MomentumLinear Momentum (Navier(Navier--Stokes Equation)Stokes Equation)III.III. Inviscid FlowInviscid Flow (Bernoulli equation) and (Bernoulli equation) and Potential FlowPotential Flow (Stream (Stream

function)function)function)function)

IncompressibleIncompressible CompressibleCompressible Incompressible Incompressible –– CompressibleCompressible Inviscid Inviscid –– ViscousViscous SteadySteady UnsteadyUnsteady Mathematical equation?Mathematical equation? Steady Steady –– UnsteadyUnsteady Rotational Rotational -- IrrotationalIrrotational

2

Motion of a Fluid ElementMotion of a Fluid ElementMotion of a Fluid ElementMotion of a Fluid Element 1. 1. Fluid Fluid TranslationTranslation: The element moves from one point to another.: The element moves from one point to another. 3. 3. Fluid Fluid RotationRotation: The element rotates about any or all of the x,y,z : The element rotates about any or all of the x,y,z

axesaxes.. Fluid Fluid DeformationDeformation::

4. 4. Angular Deformation:The element’s angles between the sides Angular Deformation:The element’s angles between the sides change.change.

2. 2. Linear Deformation:The element’s sides stretch or contract.Linear Deformation:The element’s sides stretch or contract.

3

11 FluidFluid TranslationTranslation velocity and accelerationvelocity and acceleration1. 1. Fluid Fluid TranslationTranslation velocity and accelerationvelocity and acceleration

The velocity of aThe velocity of a fluidfluid particleparticle can be expressedcan be expressed The velocity of a The velocity of a fluidfluid particleparticle can be expressedcan be expressed

TheThe total accelerationtotal acceleration ofof thethe fluidfluid particleparticle is given byis given bykwjviu)t,z,y,x(VV

Velocity field The The total accelerationtotal acceleration of of the the fluid fluid particleparticle is given byis given by

dtdz

zV

dtdy

yV

dtdx

xV

tV

DtVDa

Acceleration field

wdtdz,v

dtdy,u

dtdx

dtzdtydtxtDt

zVw

yVv

xVu

tV

DtVDa

dtdtdt

tDVDa

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

Vd

4

tDtdVd

Physical SignificancePhysical SignificancePhysical SignificancePhysical Significance

tV

zVw

yVv

xVu

tDVDa

tzyxtD Total LocalLocal

Acceleration of a particle Convective

A l i

AccelerationAcceleration--> Time> Time

Acceleration-> Space

tVV)V(

tDVDa

5

ttD

Scalar ComponentScalar Component ((외울필요외울필요없음없음))Scalar ComponentScalar Component ((외울필요외울필요없음없음))uuuu

vwvvvuvaja

zuw

yuv

xuu

tuaiax

Rectangular Rectangular

wwwvwuwaka

zw

yv

xu

taja

z

y

coordinates systemcoordinates system

zyxtz

VVVVVV 2

VVVVVVVzVV

rVV

rV

rVV

tVa r

z

2rr

rr

r

C li d i lC li d i l

VVVVVVVz

VVrVVV

rV

rVV

tVa

zzzz

zr

r

Cylindrical Cylindrical coordinates systemcoordinates system

6zV

rrV

ta z

zzz

rz

z

TranslationTranslationTranslationTranslation

All points in the element have All points in the element have the same velocitythe same velocity, then the , then the element will simply element will simply translatetranslatefrom one position to another.from one position to another.

7

22 LinearLinear DeformationDeformation 1/21/22. 2. Linear Linear DeformationDeformation 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 A change in the x dimension requires a nonzero valuenonzero value of of g qg q

A yA y y/v x/u

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

y/v

8

Linear DeformationLinear Deformation 2/22/2Linear Deformation Linear Deformation 2/22/2

The change in length of the sides may produce change in volume The change in length of the sides may produce change in volume f h lf h lof the element.of the element.

The change inThe change in ))(())(( tzyxutzyxuV

xx

The rate at which the The rate at which the V is changing per unit volume V is changing per unit volume d t di td t di t // uVd 1due to gradient due to gradient u/ u/ xx

xu

dtVd

V

1

If If v/ v/ y and y and w/ w/ z are involvedz are involved as in 2as in 2--D or 3D cases,D or 3D cases,yy ,,

))(())(())(())(( tzyxzw

yv

xutyxz

zwtzxy

yvtzyx

xuVd

Volumetric dilatation rateVolumetric dilatation rate Vzw

yv

xu

dtVd

V1

Di ergence of VDi ergence of V

9

Divergence of VDivergence of VFor an For an incompressibleincompressible fluid (constant density), the volumetric dilatation rate is fluid (constant density), the volumetric dilatation rate is zerozero..

33 Angular RotationAngular Rotation 1/41/43. 3. Angular RotationAngular Rotation 1/41/4

δαδtδαω lim

0δtOA

The angular velocity (각속도) of line OA

txv

txv

x

txx

tanFor small angles

vxv

OA

u

CCWCCW((시계반대방향시계반대방향))

yu

OB

CWCW

Positive

CCWCCW

0yu

OB

10

Positive direction

Angular RotationAngular Rotation 2/42/4Angular Rotation Angular Rotation 2/42/4

TheThe rotationrotation of the element about the zof the element about the z--axis is defined as theaxis is defined as theThe The rotationrotation of the element about the zof the element about the z--axis is defined as the axis is defined as the average of the angular velocitiesaverage of the angular velocities of the two mutually of the two mutually perpendicular lines OA and OBperpendicular lines OA and OB about the zabout the z--axisaxis

uv1li1

perpendicular lines OA and OBperpendicular lines OA and OB about the zabout the z--axisaxis.y

yxttz 2

lim2 0

x+

CCWCCW((시계반대방향시계반대방향))

zv

yw

x 21

xw

zu

y 21

(( ))

kji

In vector formIn vector form11

kji zyx In vector formIn vector form


uvwuvw 111

yu

xv

xw

zu

zv

yw

zyx

1

21

21

21

kyu

xvj

xw

zui

zv

yw

21

kyu

xvj

xw

zui

zv

ywVVcurl

21

21

21

21

21

yy

DefiningDefining vorticityvorticity V

2 >> AngularAngular rotationrotationDefining Defining vorticityvorticity V 2Defining irrotationDefining irrotation 0 V

--> > Angular Angular rotationrotation

12

gg 0 V


kji

VVcurl

21

21

21

wvuzyx

111

222

kyu

xvj

xw

zui

zv

yw

21

21

21

13

VorticityVorticityVorticityVorticity

Defining Defining VorticityVorticity ζ whichζ which is a measurement of the rotation of a measurement of the rotation of a fluid elementfluid element as it moves in the flow field:as it moves in the flow field:

uvwuvw

11

VVcurl2

Vkyu

xvj

xw

zui

zv

yw

21

21

In cylindrical coordinates systemIn cylindrical coordinates system::

rzrzr

V1rV1eVVeVV1eV

zr rrre

rze

zreV

14

44 Angular DeformationAngular Deformation 1/21/24. 4. Angular DeformationAngular Deformation 1/21/2

Angular deformationAngular deformation of a particle is given by the of a particle is given by the sum of the two sum of the two angular deformationangular deformation

vvuu

txxvtvtx

xvvty

yututy

yuu

y/x/ ξ（Xi）η（Eta）y/x/ ξ（Xi）η（Eta）

Rate of shear strainRate of shear strain or the rate of angular deformationor the rate of angular deformation

uvt

yut

xv

limlim

15

yxtt tt

...limlim00

Angular DeformationAngular Deformation 2/22/2Angular Deformation Angular Deformation 2/22/2

The rate of angular deformation in xy plane

uv

The rate of angular deformation in yz plane

yx

zv

yw

The rate of angular deformation in zx plane

uw

zx

16

Example 6 1 VorticityExample 6 1 VorticityExample 6.1 VorticityExample 6.1 Vorticity

F iF i di i l fl fi ld hdi i l fl fi ld h l i i i bl i i i b For a certain twoFor a certain two--dimensional flow field thedimensional flow field the velocity is given by velocity is given by

j)yx(2ixy4V 22

Is this flow irrotational? Is this flow irrotational?

j)y(y

17

Example 6 1Example 6 1 SolutionSolutionExample 6.1 Example 6.1 SolutionSolution

0wyxvxy4u 22

0zv

yw

21

x

0xw

zu

21

y

This flow is irrotationalThis flow is irrotational

0yu

xv

21

z

18

Conservation EquationsConservation EquationsConservation EquationsConservation Equations Continuity equationContinuity equation

Conservation of MassConservation of Mass Momentum equation (NavierMomentum equation (Navier--Stokes Eq.)Stokes Eq.)

C ti f Li M tC ti f Li M t Conservation of Linear MomentumConservation of Linear Momentum Angular momentum equationAngular momentum equation

Conservation of Angular MomentumConservation of Angular Momentum Conservation of Angular MomentumConservation of Angular Momentum Energy equationEnergy equation

Conservation of EnergyConservation of Energy

RepresentationRepresentation Integral (control volume) representationIntegral (control volume) representationDifferential representationDifferential representation

19

Conservation of MassConservation of Mass 1/51/5Conservation of Mass Conservation of Mass 1/51/5

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 The derivation is carried out by applying conservation of applying conservation of mass to a differential control volumemass to a differential control volume..

With the With the control volume representationcontrol volume representation of the conservation of massof the conservation of mass 0

CS

dAnVVdt CV

ThTh diff i l fdiff i l f f i i i ???f i i i ???The The differential formdifferential form of continuity equation???of continuity equation???

20


0CS

dAnVVdt CV

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

differential control volume

zyxt

Vdt CV

xuu|u

xuu|u Taylor series

surfacerighton theViViVn

2x

u|u2

dxx

2x

u|u2xx

n

Taylor series

21surfaceleft on the

surfaceright on the

ViViVn

ViViVn

n

V


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

uxuxu zyxxuzy

2x

xuuzy

2x

xuu

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

zyxyv

N fN fy

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

zyxzw

22


Net rate of mass wvu Net rate of mass Outflow

zyxzw

yv

xu

The differential equation for Continuity equationThe differential equation for Continuity equation 0

Vwvu

tzyxt

0

dAnVVd

zyxt

Vdt CV

CS t CV

zyxzw

yv

xudAnV

CS

23


Incompressible fluidIncompressible fluid ((density is constant and uniform)density is constant and uniform)

wvu 0

V

zw

yv

xu

Steady flowSteady flow

0)()()(0)(

V

zw

yv

xu

t

24

Example 6 2 Continuity EquationExample 6 2 Continuity EquationExample 6.2 Continuity EquationExample 6.2 Continuity Equation

Th l i f i i ibl d flTh l i f i i ibl d fl The velocity components for a certain incompressible, steady flow The velocity components for a certain incompressible, steady flow field arefield are

zyzxyvzyxu 222

Determine the form of the z component, w, required to satisfy the Determine the form of the z component, w, required to satisfy the

?w

p , , q yp , , q ycontinuity equation.continuity equation.

25

Example 6 2Example 6 2 SolutionSolutionExample 6.2 Example 6.2 SolutionSolution0wvu

The continuity equationThe continuity equation 0

zyx

The continuity equationThe continuity equation

z2u

zxv

z2x

3)(2w

zxy

)(fz3

zx3)zx(x2z

2

)y,x(f2zxz3w

26

Conservation of Linear MomentumConservation of Linear MomentumConservation of Linear MomentumConservation of Linear Momentum

Applying Newton’s second law to control volumeApplying Newton’s second law to control volume

PDF

VdVdVP

SYSDt

F VdVdmVP)system(V)system(Msystem

zVw

yVv

xVu

tVm

tDmVDF

amDt

VDm

y

Newton’s 2nd lawDt

For a For a infinitesimal system of mass dminfinitesimal system of mass dm, what’s the , what’s the tthe he 27

f y ff y f ,,differential form of linear momentum equationdifferential form of linear momentum equation??

Forces Acting on ElementForces Acting on Element 1/21/2Forces 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) forcesforcestangentialtangential (shear) forcesforces.

FFF BS

Surface 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 shear stressesof normal and shear stresses

kFjFiF szsysx

of normal and shear stresses.of normal and shear stresses.

kFjFiF bzbybx

AFn

n

limAF

11 lim

AF

22 lim

28Atn 0 At 01 At 02

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

zxyxxxF

zyyyxy

zxyxxxsx

F

zyxzyx

F

zzyzxz

zyyyxysy

F

zyxzyx

F

xbx

zzyzxzsz

zyxgF

zyxzyx

F

zbz

yby

zyxgF

zyxgF

Equation of MotionEquation of Motion

yyyy

xxxx

pp

29zzzz p

Double Subscript Notation for StressesDouble Subscript Notation for StressesDouble Subscript Notation for StressesDouble Subscript Notation for Stresses

The The directiondirection of the stressof the stress

xy

The direction of the The direction of the normal to thenormal to the planeplanenormal to the normal to the planeplaneon which the stress on which the stress actsacts

30

Equation of MotionEquation of MotionEquation of MotionEquation of Motionzzyyxx maFmaFmaF

uuuuzxyxxx

General equation of motionGeneral equation of motionyy

vwvvvuvg

zw

yv

xu

tzyxg

zyyyxy

zxyxxx

wwwvwuwg

zw

yv

xu

tzyxg

zzyzxz

y

z

wy

vx

utzyx

gz

These are the differential equations of motion for anyThese are the differential equations of motion for any fluid.fluid. How to solve u,v,w ?How to solve u,v,w ?--> > These can’t be solved because of more variables than equations, These can’t be solved because of more variables than equations, which requires more equations which requires more equations called “constitutive equations” called “constitutive equations” to solve the equations in the case of “Newtonian fluids”to solve the equations in the case of “Newtonian fluids”

31

to solve the equations in the case of Newtonian fluidsto solve the equations in the case of Newtonian fluids

StressStress--Deformation RelationshipDeformation Relationship::constitutive equations 1/21/2

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

v

Vxupp xxxx

2322

pressure fieldpressure field..

Vwpp

Vyvpp yyyy

22

322

yu

xv

Vz

pp

yxxy

zzzz

3

2Cartesian coordinates in Newtonian and C ibl fl id

zu

xw

yx

zxxz

Compressible fluids

zv

yw

zyyz

32 zzyyxxp 31

StressStress--Deformation RelationshipDeformation Relationship::constitutive equations 22/2/2

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

vxupp xxxx 2

pressure fieldpressure field..

wpp

yvpp yyyy

2

2

yu

xv

zpp

yxxy

zzzz

2Cartesian coordinates in Newtonian and I ibl fl id

zu

xw

yx

zxxz

Incompressible fluids

zv

yw

zyyz

33 zzyyxxp 31

The NavierThe Navier Stokes EquationsStokes Equations 11//22The NavierThe Navier--Stokes Equations Stokes Equations 11//22

These obtained equations of motion are called the NavierThese obtained equations of motion are called the Navier--qqStokes Equations.Stokes Equations.

UnderUnder incompressibleincompressible NewtonianNewtonian flfluidsuids the Navierthe Navier--UnderUnder incompressible incompressible Newtonian Newtonian flfluidsuids, , the Navierthe NavierStokes equations are reduced to:Stokes equations are reduced to:

222

222

2

2

2

2

2

2

zu

yu

xug

xp

zuw

yuv

xuu

tu

x

222

2

2

2

2

2

2

zv

yv

xvg

yp

zvw

yvv

xvu

tv

y

2

2

2

2

2

2

zw

yw

xwg

zp

zww

ywv

xwu

tw

z

34

The NavierThe Navier Stokes EquationsStokes Equations 22//22The NavierThe Navier--Stokes Equations Stokes Equations 22//22

The NavierThe Navier--Stokes equations apply to Stokes equations apply to both laminar and both laminar and turbulent flowturbulent flow, but , but for turbulent flowfor turbulent flow each velocity 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.

35

Laminar or Turbulent FlowLaminar or Turbulent Flow 1/21/2Laminar or Turbulent Flow Laminar or Turbulent Flow 1/21/2

The flow of a fluid in a pipe may be The flow of a fluid in a pipe may be Laminar ? Or Laminar ? Or Turbulent ?Turbulent ?

Osborne ReynoldsOsborne Reynolds, a British scientist and mathematician, , a British scientist and mathematician, was the first to distinguish the difference between these was the first to distinguish the difference between these ggclassification of flow by using a classification of flow by using a simple apparatussimple apparatus as as shown.shown.

36

Laminar or Turbulent FlowLaminar or Turbulent Flow 2/22/2Laminar or Turbulent Flow Laminar or Turbulent Flow 2/22/2

For “For “small enough flowratesmall enough flowrate” the dye streak will remain as a ” the dye streak will remain as a wellwell--defined linedefined line as it flows along, with only slight blurring due as it flows along, with only slight blurring due t l l diff i f th d i t th di tt l l diff i f th d i t th di tto molecular diffusion of the dye into the surrounding water.to molecular diffusion of the dye into the surrounding water.

For a somewhat larger “For a somewhat larger “intermediate flowrateintermediate flowrate” the dye ” the dye fl i i d d i i b f i lfl i i d d i i b f i lfluctuates in time and space, and intermittent bursts of irregular fluctuates in time and space, and intermittent bursts of irregular behavior appear along the streak.behavior appear along the streak.

F “F “l h fl tl h fl t ” h d k l” h d k lFor “For “large enough flowratelarge enough flowrate” the dye streak almost ” the dye streak almost immediately become blurred and spreads across the entire pipe in immediately become blurred and spreads across the entire pipe in aa randomrandom fashionfashiona a randomrandom fashion.fashion.

37

Time Dependence of Time Dependence of Fluid Velocity at a PointFluid Velocity at a Point

38

Indication of Indication of Laminar or Turbulent FlowLaminar or Turbulent Flow ThTh fl tfl t h ld bh ld b l d b R ldl d b R ld The term The term flowrateflowrate should be should be replaced by Reynolds replaced by Reynolds

numbernumber, ,where , ,where VV is the average velocity in the pipe, is the average velocity in the pipe, and and L L is the characteristic dimension of a flow. is the characteristic dimension of a flow. LL is usually is usually D D

/VLRe yy

(diameter)(diameter) in a pipe flow. in a pipe flow. --> a measure of inertial force to the > a measure of inertial force to the viscous force.viscous force.

I iI i l h fl id l il h fl id l i h d i h h f hh d i h h f h It is It is not only the fluid velocitynot only the fluid velocity that determines the character of the that determines the character of the flow flow –– its density, viscosity, and the pipe size are of equal its density, viscosity, and the pipe size are of equal importance.importance.pp

For general engineering purpose, the flow in a For general engineering purpose, the flow in a round piperound pipeLaminarLaminar 2100R e TransitionalTransitionalTurbulentTurbulent

2100R e

4000＞R e39

4000＞R e

Some Simple Solutions for Viscous, Some Simple Solutions for Viscous, Incompressible FluidsIncompressible FluidsA principal difficulty in solving the NavierA principal difficulty in solving the Navier--Stokes Stokes

equations is because of their equations is because of their nonlinearitynonlinearity arising from the arising from the convective acceleration termsconvective acceleration terms..

There are no general analytical schemes for solving There are no general analytical schemes for solving g y gg y gnonlinear partial differential equations.nonlinear partial differential equations.

There are aThere are a few special cases for which the convectivefew special cases for which the convectiveThere are a There are a few special cases for which the convective 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.often possible.often possible.

40

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 1/1/44

1.1. Schematic:Schematic:2.2. Assumptions: Incompressible, Newtonian, Steady, One dimensional flowAssumptions: Incompressible, Newtonian, Steady, One dimensional flowp p , , y,p p , , y,3.3. Continuity equationContinuity equation4.4. The NavierThe Navier--Stokes equations Stokes equations

yuuxu

zw

yv

xuV

00 0

pyu

xp

0 2

2

222

2

2

2

2

2

2

vvvpvvvv

zu

yu

xug

xp

zuw

yuv

xuu

tu

x

zp

gyp

0

0

2

2

2

2

2

2

222

zw

yw

xwg

zp

zww

ywv

xwu

tw

zv

yv

xvg

yp

zvw

yvv

xvu

tv

z

y

41

z zyxzzyxt

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 2/2/44

55 Boundary conditions (B C )Boundary conditions (B C ) u=0 at y=u=0 at y= h u=0 at y=hh u=0 at y=h (no(no slipslip

5.5. Boundary conditions (B.C.) Boundary conditions (B.C.) u=0 at y=u=0 at y=--h u=0 at y=hh u=0 at y=h (no(no--slip slip boundary condition)boundary condition)

6.6. Solve the equations with B.C.Solve the equations with B.C.

0

0

p

gyp xfgyp 1 IntegratingIntegrating

2

2

0

0

yu

xpz

??21

21212 cccycy

xpu

IntegratingIntegrating

yx 2 x

212 2

1,0 hpcc

2 x

221 hypu

42

2

hyx

u

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 33//44

Shear stress distributionShear stress distributionShear stress distributionShear stress distribution

yxp

yu

yx

Volume flow rate Volume flow rate per unit depth (z direction)per unit depth (z direction)

xy

xphdyhy

xpudyq

h

h

h

h 32)(

21 3

22

21

12

12 0constant pppxx

ppxp

33

21

3

122

pressureoutlet theis and pressureinlet theis where,3

2

hph

ppphq

433resistanceFlow 1

32

32

hV

Riphqphq

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 44//44

Average velocityAverage velocity per unit depthper unit depth2 phqV

Point of maximum velocityPoint of maximum velocity32p

hqVaverage

Point of maximum velocityPoint of maximum velocity

0du at y=00

dy at y 0

VphU 32

averageVxpUuu

22max

44

Couette FlowCouette Flow 1/31/3 (HW)(HW)Couette Flow Couette Flow 1/31/3 (HW)(HW)

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 “both of the “both plates stationary” case.plates stationary” case.

45

Couette FlowCouette Flow 2/32/3Couette Flow Couette Flow 2/32/3

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

??12121

2 cccycypu

u=U at y=bu=U at y=b

1 pU

??2 2121 cccycy

xu

021

21

cbxp

bUc

Velocity distributionVelocity distribution

byxpy

xp

bUyu

21

21 2

yb

by

xp

Ub

by

Uu 1

2

2

xp

UbP2

2

46

xU2

Couette FlowCouette Flow 3/33/3Couette Flow Couette Flow 3/33/3

ii rrbrU Simplest type of Couette flow

yp)/( ioi

ioi

rrrrrbrU

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 velocityconstant angular velocity.

Flow in the narrow gap of a journal bearing.

47

j g

Steady, Laminar Flow Steady, Laminar Flow (Hagen(Hagen--Poiseuille Poiseuille Flow) Flow) in Circular Tubes in Circular Tubes 1/51/5

1.1. Schematic:Schematic:2.2. Assumptions: Incompressible, Newtonian, Steady, Laminar, One dimensional Assumptions: Incompressible, Newtonian, Steady, Laminar, One dimensional

flowflowflowflow

0,0,0 zr vvv

3.3. Continuity equationContinuity equation

44 Th N iTh N i S k iS k i

rvvzv

zzz

0

4.4. The NavierThe Navier--Stokes equations Stokes equations 5.5. Boundary Conditions: Boundary Conditions: At r=0, the velocity vAt r=0, the velocity vzz is finite. At r=R, the velocity vis finite. At r=R, the velocity vzz is is

zero.zero.

486.6. Solve the equation with B.C.Solve the equation with B.C.

From the NavierFrom the Navier--Stokes EquationsStokes Equations in in Cylindrical coordinatesCylindrical coordinates General motion of an General motion of an incompressible Newtonian fluidincompressible Newtonian fluid is governed by the is governed by the

continuity equation and the momentum equationcontinuity equation and the momentum equation

Mass conservation

Navier-Stokes Equation in a cylindrical coordinatein a cylindrical coordinate

Acceleration

49

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

NavierNavier –– Stokes equation reduced toStokes equation reduced toNavier Navier –– Stokes equation reduced to Stokes equation reduced to

pg sin0 ,sin 1 zfrgp

pg

r1cos0

1fgp

zfgyp 1 IntegratingIntegrating

vrp

rg

z10

IntegratingIntegrating

r

rrrz

0 IntegratingIntegrating

1 ??ln41

21212 cccrcr

zpvz

50

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

At r 0 the elocitAt r 0 the elocit is finite At r R the elocitis finite At r R the elocitAt r=0, the velocity vAt r=0, the velocity vzz is finite. At r=R, the velocity vis finite. At r=R, the velocity vzzis zero.is zero.

221 4

1,0 Rzpcc

Velocity distributionVelocity distribution

221 Rrpv

4Rr

zvz

51

Steady Laminar Flow in Circular TubesSteady Laminar Flow in Circular Tubes 4/54/5Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 4/54/5

The shear stress distributionThe shear stress distribution

prdv z

Volume flow rateVolume flow rate

zdrrz 2

Volume flow rateVolume flow rate

zpRrdruQ

R

z 8

.....24

0 z80

pppzp

/constant 12

ppRzpRQ

z

128D

88

444

52

z 12888

Steady Laminar Flow in Circular TubesSteady Laminar Flow in Circular Tubes 5/55/5Steady, Laminar Flow in Circular Tubes Steady, Laminar Flow in Circular Tubes 5/55/5

Average velocityAverage velocity2 pRQQ 82

pRRQ

AQVaverage

Point of maximum velocityPoint of maximum velocity

0dr

dv z at r=0at r=022

max 124

Rr

vvVpRv z

average

53

max4 Rv

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

(HW)(HW)

Boundary conditionsBoundary conditions

For steady, laminar flow in For steady, laminar flow in annularannular tubestubes

Boundary conditionsBoundary conditionsvvzz = 0 , at r = r= 0 , at r = roovv = 0 at r = r= 0 at r = rvvzz = 0 , at r = r= 0 , at r = rii

54

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

Th l i di ib iTh l i di ib i

oi rrrrrpv ln1 22

22

The velocity distributionThe velocity distribution

oio

oz rrrrr

zv ln

)/ln(4

Th l f flTh l f flThe volume rate of flowThe volume rate of flow

)()2(222

44 ior rrrrpdrrvQ o

)/ln(8)2(

ioior z rr

rrz

drrvQi

The maximum velocity occurs at r=rThe maximum velocity occurs at r=rmmyy mm

2/122

io rrr0vz

55)/ln(2

io

m rrr0

r

Inviscid FlowInviscid FlowInviscid FlowInviscid Flow

Shear stresses developShear stresses develop in a moving fluid in a moving fluid because of the viscositybecause of the viscosity of of the fluid.the fluid.

F fl id h iF fl id h i th i it i llth i it i ll dd For some common fluid, such as air, For some common fluid, such as air, the viscosity is smallthe viscosity is small, and , and therefore it therefore it seems reasonable to assume that under some seems reasonable to assume that under some circumstances we may be able to simply neglect the effect ofcircumstances we may be able to simply neglect the effect ofcircumstances we may be able to simply neglect the effect of circumstances we may be able to simply neglect the effect of viscosityviscosity..

Flow fields in which the shear stresses are assumed to be negligible Flow fields in which the shear stresses are assumed to be negligible g gg gare said to be inviscid, or frictionlessare said to be inviscid, or frictionless..

D fi th th ti f th l tD fi th th ti f th l t

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

56

Euler’s Equation of MotionEuler’s Equation of MotionEuler s Equation of MotionEuler s Equation of Motion

UnderUnder inviscid flows: frictionless conditioninviscid flows: frictionless condition, , the the equations of motion are reduced toequations of motion are reduced to Euler’s EquationEuler’s Equation::

xgxp

zuw

yuv

xuu

tu

ygyp

zvw

yvv

xvu

tv

Euler’s EquationEuler’s Equation

zgzp

zww

ywv

xwu

tw

yy

zzyxt

pgVD

57

pgDt

Bernoulli EquationBernoulli Equation 1/31/3Bernoulli Equation Bernoulli Equation 1/31/3

Euler’s equation for Euler’s equation for steadysteady flowflow along a streamlinealong a streamline isis

V)V(pg

V)V(pg Selecting the coordinate system with the zSelecting the coordinate system with the z--axis vertical so that axis vertical so that

zgg

the acceleration of gravity vector can be expressed asthe acceleration of gravity vector can be expressed as

VVVV21VV

gg

Vector identity ….Vector identity ….2

VVVVpzg

)(2

58

2


p 21

1

V perpendicular to perpendicular to V

VVVzgVp 2

21

sd sdVVsdzgsdVsdp

2

21

sd

With With kdzjdyidxsd

tangential vector on a streamlinetangential vector on a streamline

dVdVdVdVkdjdidkVjViVdV

dpdzzpdy

ypdx

xpkdzjdyidxk

zpj

ypi

xpsdp

2222222

2 1111

gdzdzzdyzdxzgkdzjdyidxkzjzizgsdzg

dVdzz

Vdyy

Vdxx

Vkdzjdyidxkz

Vjy

Vix

VsdV

22

2222

59

gz

yyx

gjyz

jyx

gg


1

1d

021 2

sdzgsdVsdp

021 2 gdzVddp

constant2

2

gzVdp

Integrating …Integrating …2

For For steadysteady,, inviscid, incompressible fluidinviscid, incompressible fluid (commonly called ideal (commonly called ideal fluids)fluids) along a streamlinealong a streamline Bernoulli equation is given byBernoulli equation is given by

constant2

gzVp

fluids) fluids) along a streamlinealong a streamline Bernoulli equation is given byBernoulli equation is given by

Bernoulli equationBernoulli equation60

2g

Bernoulli equationBernoulli equation

Irrotational FlowIrrotational Flow 1/1/22Irrotational Flow Irrotational Flow 1/1/22

Irrotation ? The irrotational condition is

0V

In rectangular coordinates system

0V In rectangular coordinates system

0wuvwuv

In cylindrical coordinates system

0xzzyyx

0Vr1

rrV

r1

rV

zV

zVV

r1 rzrz

61

Irrotational FlowIrrotational Flow 2/2/22Irrotational Flow Irrotational Flow 2/2/22

A general flow field would not be irrotational flow.A general flow field would not be irrotational flow.A special uniform flow field is an example of an A special uniform flow field is an example of an p pp p

irrotationirrotationalal flowflow

62

Bernoulli Equation for Irrotational FlowBernoulli Equation for Irrotational Flow 1/31/3Bernoulli Equation for Irrotational Flow Bernoulli Equation for Irrotational Flow 1/31/3

The Bernoulli equation forThe Bernoulli equation for steady, incompressible, and inviscid steady, incompressible, and inviscid flowflow isis

2Vp

Th i b li d bTh i b li d b i hi h

constant2

gzVp

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 streamlinestreamline to streamlinestreamline to streamline..

Under additionalUnder additional irrotational conditionirrotational condition, , the Bernoulli equation ?the Bernoulli equation ?Starting with Euler’s equation in vector formStarting with Euler’s equation in vector form

VVVpg

)(Starting with Euler s equation in vector formStarting with Euler s equation in vector form

VVVV21kgp1V)V(

VVt

pg )(

63

2

ZERO Regardless of the direction of dsZERO Regardless of the direction of ds


With irrotaionalirrotaional condition 0V

VVVVkgpVV

11)(

kVVV

111 2 d

VVVVkgpVV 2

)(

kgpVVV 22

2 rd11 kdzjdyidxrd

121 2 rdkgrdprdV

kdzjdyidxrd

Not a streamline

021

21 22 gdzVddpgdzdpVd

64


Integrating for incompressible flowIntegrating for incompressible flow

constant2

gzVpcontant

2

gzVdp

Thi i i lid bThi i i lid b i i di i d

constant2

gz

contant2

gz

This equation is valid between This equation is valid between any two points in a steady, any two points in a steady, incompressible, inviscid, and irrotational flowincompressible, inviscid, and irrotational flow irrespective of irrespective of streamlinesstreamlinesstreamlinesstreamlines..

2

222

1

211 z

2Vpz

2Vp

21 g2g2

65

Stream FunctionStream Function 1/61/6Stream Function Stream Function 1/61/6

StreamlinesStreamlines:: Lines tangent to the instantaneous velocity vectors at Lines tangent to the instantaneous velocity vectors at every point.every point.

St f ti ΨSt f ti Ψ( )( ) [P i] ? U d t t th l it[P i] ? U d t t th l it Stream function ΨStream function Ψ(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 component u(x,y,t) and v(x,y,t) of a ““twotwo--dimensionaldimensionalincompressibleincompressible”” flowflowincompressibleincompressible flow.flow.

Define a function ΨDefine a function Ψ(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 asy g gy g g

xv

yu

66


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

0xyyx

0yv

xu 22

ΨΨ(x,y) (x,y) isis sstill unknown for a particular problem, but at least we have till unknown for a particular problem, but at least we have i lif th l ii lif th l i b h i t d t ib h i t d t i l kl ksimplify the analysissimplify the analysis by having to determine by having to determine only one unknownonly one unknown, , ΨΨ(x,y)(x,y) , rather than the two , rather than the two unknown unknown function u(x,y) and v(x,y).function u(x,y) and v(x,y).

67


Another advantage of using stream function is related to the fact that Another advantage of using stream function is related to the fact that line along which line along which ΨΨ(x,y) =constant(x,y) =constant are streamlines.are streamlines.

H t ? F th d fi iti f th t li th t th lH t ? F th d fi iti f th t li th t th l How to prove ? From the definition of the streamline that the slope How to prove ? From the definition of the streamline that the slope at any point along a streamline is given byat any point along a streamline is given by

uv

dxdy

streamline

streamline

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

68

Ve oc ty a d ve oc ty co po e t a o g a st ea eVe oc ty a d ve oc ty co po e t a o g a st ea e


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

udyvdxdyy

dxx

d

0udyvdx0d

d

Along a line of constant ΨAlong a line of constant Ψ

uv

dxdy

streamline

This is the definition for a streamline. Thus, This is the definition for a streamline. Thus, if we know the if we know the stream stream functionfunction Ψ(x,y) we Ψ(x,y) we can can plot lines of constantplot lines of constant Ψto provide the family of Ψto provide the family of streamlines that are helpful in streamlines that are helpful in visualizing the pattern of flowvisualizing the pattern of flow. There are an infinite number of streamlines that make up a . There are an infinite number of streamlines that make up a

69

particular flow field, since for each constant value assigned to Ψa streamline can be drawn.particular flow field, since for each constant value assigned to Ψa streamline can be drawn.


The actual numerical value associated with a particular streamline is The actual numerical value associated with a particular streamline is not of particular significance, but the change in the value of Ψnot of particular significance, but the change in the value of Ψ is is related to the volume rate of flowrelated to the volume rate of flowrelated to the volume rate of flow.related to the volume rate of flow.

dq : dq : volume rate of flow passing between the two streamlinesvolume rate of flow passing between the two streamlines. Flow . Flow never crosses streamlines by definitionnever crosses streamlines by definitionnever crosses streamlines by definition. never crosses streamlines by definition.

ddxx

dyy

vdxudydq

rightleft tofromisflowthe0If

122

1

q

dq

left. right to from is flow the,0 Ifright.left tofromis flow the,0 If

qq

70


Thus the Thus the volume flow ratevolume flow rate between any two streamlines can be between any two streamlines can be written as written as the difference between the constant values of Ψthe difference between the constant values of Ψ defining defining two streamlinestwo streamlinestwo streamlines.two streamlines.

The velocity will be relatively high wherever the streamlines are The velocity will be relatively high wherever the streamlines are close together and relatively low wherever the streamlines are farclose together and relatively low wherever the streamlines are farclose together, and relatively low wherever the streamlines are far close together, and relatively low wherever the streamlines are far apart.apart.

71

Example 6 3 Stream FunctionExample 6 3 Stream FunctionExample 6.3 Stream FunctionExample 6.3 Stream Function

Th l i i d i iblTh l i i d i ibl 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 sketchDetermine the corresponding stream function and show on a sketch

4xv2yu

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

72

Example 6 3Example 6 3 SolutionSolutionExample 6.3 Example 6.3 SolutionSolutionFrom the definition of the stream functionFrom the definition of the stream functionFrom the definition of the stream functionFrom the definition of the stream function

x4vy2u

22 yx2 Ψ=0Ψ=0

(y)fx2(x)fy 22

12

xy

y

(y)( )y 21

Cyx2 22

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

ΨΨ≠≠001xy 22

73

12/

Velocity PotentialVelocity Potential Φ(Φ(x y z t)x y z t) 1/1/33Velocity Potential Velocity Potential Φ(Φ(x,y,z,t)x,y,z,t) 1/1/33

Th f i fTh f i f di i l i ibldi i l i iblThe stream function for The stream function for twotwo--dimensional incompressible dimensional incompressible flowflow isis ΨΨ(x,y) (x,y)

F i i l fl h l i bF i i l fl h l i bFor an irrotational flow, the velocity components can be For an irrotational flow, the velocity components can be expressed in terms of a scalar function expressed in terms of a scalar function Φ(Φ(x,y,z,t)x,y,z,t) asas

zw

yv

xu

where where Φ(Φ(x,y,z,t)x,y,z,t) is called the is called the velocity potentialvelocity potential..

zyx

VV

0

74


In vector formIn vector formV

For an incompressible flow For an incompressible flow V

0V Also Also called a called a potential flowpotential flow

0V

ForFor incompressible irrotational flowincompressible irrotational flow

0VV 2

2

2

2

2

22

For For incompressible, irrotational flowincompressible, irrotational flow

zyx 222 Laplace’s equation

75Laplacian operatorLaplacian operator


Inviscid, incompressible, irrotational fields are governed Inviscid, incompressible, irrotational fields are governed by Laplace’s equationby Laplace’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 givenTo complete the mathematical formulation of a givenTo 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 theare usually velocities specified on the boundaries of theare 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.

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FUNDAMENTALS OF FLUID MECHANICSFLUID ...cau.ac.kr/~jjang14/FME/Chap6.pdfChapter 6 Flow AnalysisChapter 6 Flow Analysis Using Differential MethodsDifferential Methods 1 MAIN TOPICSMAIN

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