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1

Computational Fluid Dynamics

An Introduction

SimulationinComputerGraphicsUniversityofFreiburg

WS04/05

UniversityofFreiburg-InstituteofComputerScience- ComputerGraphicsLaboratory

Thisslidesetisbasedon:

n JohnD.Anderson,Jr.,ComputationalFluidDynamicsTheBasicswithApplications, McGraw-Hill,Inc.,

NewYork,ISBN0-07-001685-2

Acknowledgement


Motivation

D.Enright,S.Marschner,R.Fedkiw,

AnimationandRenderingof

ComplexWaterSurfaces,

Siggraph 2002,ACMTOG,

vol.21,pp.736-744,2002.


n introduction

n pre-requisites

n governingequations

n continuityequation

n momentumequation

n summary

n solutiontechniquesn Lax-Wendroff

n MacCormack

n numericalaspects

n recentresearchingraphics

Outline


2/15

2


n physicalaspectsoffluidflowaregovernedby

threeprinciples

n massisconservedn force=mass acceleration(Newtonssecondlaw)n energyisconserved(notconsideredinthislecture)

n principlescanbedescribedwithintegralequationsorpartialdifferentialequations

n inCFD,thesegoverningequationsarereplacedby

discretized algebraicforms

n algebraicformsprovidequantitiesatdiscretepointsintimeandspace,noclosed-formanalyticalsolution

Introduction


NumericalSolution- Overview

GoverningEquations

DiscretizationSystem ofAlgebraicEquations

EquationSolver

Approx.Solution

Navier-StokesEuler

FiniteDifference

DiscreteNodalValues

Lax-Wendroff

MacCormack


n x,y,z 3Dcoordinatesystem

n t time

n ( x,y,z,t ) density

n v ( x,y,z,t ) velocity

n v ( x,y,z,t )=(u ( x,y,z,t ),v ( x,y,z,t ),w ( x,y,z,t ))T

Continuous Quantities


Problem

v (x,y,z,t)

(x,y,z,t)

v (x,y,z,t+t) (x,y,z,t+t)


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n introduction

n pre-requisites

n governingequationsn continuityequation

n momentumequation

n summary

n solutiontechniques

n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n infinitesimallysmallfluidelementmovingwiththeflow

n ( x1,y1,z1) positionattimet1

n ( x2,y2,z2) positionattimet2

n v1(x1,y1,z1,t1)=(u(x1,y1,z1,t1 ),v (x1,y1,z1,t1 ),w (x1,y1,z1,t1 ))T

n v2(x2,y2,z2,t2)=(u(x2,y2,z2,t2 ),v (x2,y2,z2,t2 ),w (x2,y2,z2,t2 ))T

n 1 ( x1,y1,z1,t1 )

n 2 ( x2,y2,z2,t2 )

Substantial Derivative of


n Taylorseriesatpoint1

t2 t1

Substantial Derivative of

( ) ( ) ( ) ( )12

1

12

1

12

1

12

1

12tt

tzz

zyy

yxx

x

+

+

+

+=

112

12

112

12

112

12

112

12

+

+

+

=

ttt

zz

ztt

yy

ytt

xx

xtt

111112

12

+

+

+

=

tw

zv

yu

xtt


n substantialderivativeof

n substantialderivative=localderivative+convectivederivative

n localderivative timerateofchangeatafixedlocation

n convectivederivative timerateofchangeduetofluidflow

n subst.derivative=totalderivativewithrespecttotime

Substantial Derivative

tzw

yv

xu

Dt

D

+

+

+

T

zyxtDt

D

=+

,,)(v


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4


n divergenceofvelocityv =timerateofchangeofthe

volumeVofamovingfluidelementperunitvolume

Divergence ofv

( )Dt

VD

Vz

w

y

v

x

u

1=+

+

= v


n introduction

n pre-requisites


n momentumequation

n summary


n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n massisconserved

n infinitesimallysmallfluidelementmovingwiththeflow

n fixedmassm,variablevolumeV, variabledensity

n timerateofchangeofthemassofthemovingfluid

elementiszero

Continuity Equation

Vm =

( ) 0=+==Dt

VD

Dt

DVDt

VD

Dt

mD

01 =+

Dt

VD

VDt

D

0=+ v

Dt

D


Continuity Equation

0=+ v

Dt

D

divergenceofvelocity-

timerateofchangeofvolume

ofamovingfluidelementper

volume

substantialderivative

timerateofchangeof

densityofamoving

fluidelement


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n non-conservationform(considersmovingelement)

n manipulation

n conservationform(considerselementatfixedlocation)

Continuity Equation

0=+ v

Dt

D

( )vvvv

+

=++

=+ttDt

D

( ) 0=+

v

t


n motivationforconservationform

n infinitesimallysmallelementatafixedlocation

n

massfluxthroughelementn differenceofmassinflowandoutflow=netmassflow

n netmassflow=timerateofmassincrease

Continuity Equation

( ) 0=+

v

t

netmassflow

pervolume

timerateofmass

increasepervolume


n introduction

n pre-requisites



n momentumequation

n summary

n

solutiontechniquesn Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n consideramovingfluidelement

n physicalprinciple:F= ma (Newtonssecondlaw)n twosourcesofforces

n bodyforces

n gravity

n surfaceforces

n

basedonpressuredistributiononthesurfacen basedonshearandnormalstressdistributiononthesurface

duetodeformationofthefluidelement

normal

stress

Momentum Equation

pressurefluid

element

gravity

shear

stress

friction

velocity

velocity


6/15


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(yx + yx /ydy)dx dz

Surface Forces in x

dxdy

dzpdy dz (p+p/xdx) dydz

xxdy dz (

xx+

xx/xdx)dy dz

yx dxdz zx notillustrated

dzdydxx

dzdydxx

pppF

xx

xx

xxx

++

+=

dzdydzz

dzdxdyy

zx

zx

zxyx

yx

yx

++

++

dzdydxzyxx

pF zx

yxxxx

+

+

+

=


Momentum Equation in x

dzdydxfdzdydxzyxx

p

Dt

Du

dzdydx xzxyxxx

+

+

+

+

=

xxFam =

mass acceleration

timerateofchange

ofvelocityofa

movingfluidelement

surfaceforce

pressure

normalstress

shearstress

bodyforce

gravity

xzxyxxx fzyxx

p

Dt

Du

+

+

+

+

=


n viscousflow,non-conversationform

n Navier-Stokesequations

Momentum Equation

x

zxyxxx fzyxx

p

Dt

Du

+

+

+

+

=

y

zyyyxyf

zyxy

p

Dt

Dv

+

+

+

+

=

z

zzyzxz fzyxz

p

Dt

Dw

+

+

+

+

=


n obtainingtheconservationformofNavier-Stokes


ut

u

Dt

Du += v

( )t

ut

u

t

u

+

=

( )

t

u

tu

t

u

=

( ) ( ) ( ) uuu += vvv ( ) ( ) ( )vvv = uuu

( )( ) ( )vv u

tu

t

u

Dt

Du

+

+

=

continuityequation

( )( )vu

t

u

Dt

Du

+

=


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n Navier-Stokesequationsinconservationform

Momentum Equation

( ) ( ) xzxyxxx fzyxx

putu ++++=+

v

( )( )

y

zyyyxyf

zyxy

pv

t

v

+

+

+

+

=+

v

( )( )

z

zzyzxz fzyxz

pw

t

w

+

+

+

+

=+

v


n Newton:shearstressinafluidisproportional

tovelocitygradients

n mostfluidscanbeassumedtobenewtonian,butbloodisapopularnon-newtonianfluid

n Stokes:

n isthemolecularviscositycoefficientn isthesecondviscositycoefficientwith =-2/3

Newtonian Fluids

( )x

uxx

+= 2v ( )

y

vyy

+= 2v ( )

z

wzz

+= 2v

+

==y

u

x

vyxxy

+

==x

w

z

uzxxz

+

==z

v

y

wzyyz


n intermolecularforcesarenegligible

n R specificgasconstant

n T temperature

n termal equationofstate

Perfect Gas

TRp =


n now,p canbeexpressedusingdensityand

canbeexpressedusingvelocitygradients,

fx isuser-defined(e.g.gravity)


( )( )

x

zxyxxx fzyxx

pu

t

u

+

+

+

+

=+

v


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n introduction

n pre-requisites


n momentumequation

n summary


n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n thus,fourequationsandfourunknowns ,u,v,w

Governing Equations Viscous Compressible Flow

( )( )

x

zxyxxx fzyxx

pu

t

u

+

+

+

+

=+

v

( )( ) y

zyyyxyf

zyxy

pv

t

v

+

+

+

+

=+

v

( )( )

z

zzyzxz fzyxz

pw

t

w

+

+

+

+

=+

v

( ) 0=+

v

t

continuityequation

momentume

quation


n Eulerequations

Governing Equations Inviscid Compressible Flow

( )( )

xf

x

pu

t

u

+=+

v

( ) ( )y

fy

pv

t

v

+=+

v

( )( )

zf

z

pw

t

w

+

=+

v

( ) 0=+

v

tcontinuityequation

momentumeq

uation


n coupledsystemofnonlinearpartialdifferentialequations

n normalandshearstresstermsarefunctionsofthevelocitygradients

n pressureisafunctionofthedensity

n onlymomentumequationsareNavier-Stokesequations,

howeverthenameiscommonlyusedforthefullset

ofgoverningequations(plusenergyequation)

Commentson the Governing Equations


10/15

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n canbeobtaineddirectly fromacontrolvolumefixedin

space

n

considerflux ofmass,momentum(andenergy)intoandoutofthevolume

n haveadivergencetermwhichinvolvestheflux ofmass(v),momentuminx,y,z(uv,vv,wv)

n canbeexpressedinagenericform

Governing Equations inConservation Form

JHGFU =

+

+

+

zyxt


n viscousflow

n inviscid flow

Generic Form

=

w

v

u

U

+=

xz

xy

xx

uw

uv

puu

u

F

+ =

yz

yy

yx

tvw?

tpvv?

tvu?

v?

G

=

z

y

x

f?

f?

f?

0

J

++ =

zz

zy

zx

tpww?

tpwv?

twu?

w?

H

=

w

v

u

U

+

=

uw

uv

puu

u

F

+=

vw?

pvv?

vu?

v?

G

=

z

y

x

f?

f?

f?

0

J

++

=

pww?

pwv?

wu?

w?

H


n U solutionvector(tobesolvedfor)

n F,G,H fluxvectors

n J sourcevector(externalforces,energychanges)

n dependentflowfieldvariablescanbesolvedprogressively

instepsoftime(time-marching )n spatialderivativesareknowfromprevioustimesteps

n numberscanbeobtainedfordensity andfluxvariablesu,v,w

Generic Form

zyxt

+

+

= HGF

JU


n inmanyreal-worldproblems,discontinuouschangesin

theflow-fieldvariables,v occur(shocks,shockwaves)n problem:differentialformofthegoverningequationsassumesdifferentiable(continuous)flowproperties

n simple1-Dshockwave:1 u1 =2 u2

n conservationformsolvesforu,seesnodiscontinuityn positiveeffectonnumericalaccuracyandstability

Motivation for Different Formsof the Governing Equations

x

x

u

x

u


11/15

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n introduction

n pre-requisites


n momentumequation

n summary


n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n flow-fieldvariables,u,v,w areknownateachdiscretespatialposition(x,y,z )attimet

n

Lax-Wendroff computesallinformationattimet+tn second-orderaccuracyinspaceandtime

n explicit,finite-differencemethod

n particularlysuitedtomarchingsolutions

n marchingsuitedtohyperbolic,parabolicPDEs

n unsteady(time-dependent),compressibleflowisgovernedbyhyperbolicPDEs

Lax-Wendroff Technique


n 2D,inviscid flow,nobodyforce,innon-conservationform

Lax-WendroffExample

+

+

+

=

yv

y

v

xu

x

u

t

+

+

=

x

TR

y

uv

x

uu

t

u

+

+

=

y

TR

y

vv

x

vu

t

v


n ti,j densityatposition(i, j )andtimetn ti,j ,uti,j ,vti,j areknown

Taylor Series Expansions

( )...

2

2

,

2

2

,

,,+

+

+=+t

tt

t

t

ji

t

ji

t

ji

tt

ji

( )...2

2

,

2

2

,

,, +

+

+=+ t

t

utt

uuu

t

ji

t

ji

t

ji

tt

ji

( )...

2

2

,

2

2

,

,,+

+

+=+t

t

vt

t

vvv

t

ji

t

ji

t

ji

tt

ji


12/15

12


n /t canbereplacedbyspatialderivativesgiveninthegoverningequations

n usingsecond-ordercentraldifferences

Taylor Expansion - Density

( )...

2

2

,

2

2

,

,,+

+

+=+

t

tt

t

t

ji

t

ji

t

ji

tt

ji

+

+

+

=

yv

y

v

xu

x

u

t

+

+

+

=

++++

yv

y

vv

xu

x

uu

t

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji 2222

,1,1

,

,1,1

,

,1,1

,

,1,1

,

,


n 2/t2 canbeobtainedbydifferentiatingthegoverningequationwithrespecttotime

n mixedsecondderivativescanbeobtainedbydifferentiating

thegoverningequationswithrespecttoaspatialvariable,

e.g.

n bracketsmissingonpp.219,220inAndersonsbook!


+

+

+

+

+

+

+

=

t

v

ytyv

ty

v

ty

v

t

u

xtxu

tx

u

tx

u

t

2222

2

2

+

+

+

+

=

2

22

2

22

2

22

x

TR

x

TR

x

v

y

u

yx

uv

x

u

x

uu

tx

u


n 2/t2 canbecomputedusingcentraldifferencesforspatialderivatives

n higher-ordertermsarerequired,e.g.

n ... u,v arecomputedinthesameway


( )2,1,,1

,

2

2 2

x

uuu

x

ut

ji

t

ji

t

ji

t

ji

+=

+

yx

uuuu

yx

ut

jit

jit

jit

jit

ji

+=

++++4

1,11,11,11,1

,

2


n introduction

n pre-requisites



n momentumequation

n summary


n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


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n samecharacteristicslikeLax-Wendroff

n second-orderaccuracyinspaceandtime

n requiresonlyfirsttimederivativen predictor corrector

n illustratedfordensity

MacCormack Technique

tt

av

ji

t

ji

tt

ji

+=+,

,,


n predictorstepfordensity

n u,v arepredictedthesameway


+

+

+

=

++++

y

v

y

vv

x

u

x

uu

t

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji

t

jit

ji

t

ji 2222

,1,1

,

,1,1

,

,1,1

,

,1,1

,

,

tt

t

ji

t

ji

tt

ji

+=+

,

,,

tt

uuu

t

ji

t

ji

tt

ji

+=+

,

,,t

t

vvv

t

ji

t

ji

tt

ji

+=+

,

,,


n correctorstepfordensity

n correspondstoHeun schemeforODEsn otherhigher-orderschemes,e.g.Runge-Kutta 2or4,couldbeusedaswell


+

+

+

=

+

+++

+

+++

+

+++

+

+++

+

yv

y

vv

xu

x

uu

t

ttji

ttjitt

ji

ttji

ttjitt

ji

ttji

ttjitt

ji

ttji

ttjitt

ji

tt

ji2222

,1,1

,

,1,1

,

,1,1

,

,1,1

,

,

ttt

tt

tt

ji

t

ji

t

ji

av

ji

t

ji

tt

ji

+

+=

+=+

+

,,

,

,

,,2

1


n Lax-Wendroff andMacCormack canbeusedfor

unsteadyflow

n non-conservationform

n conservationform

n viscousflow

n inviscid flow

n higher-orderaccuracyrequiredtoavoidnumerical

dissipation,artificialviscosity,numericaldispersionn note,thatviscosityisrepresentedbysecondpartialderivativesinthegoverningequations

Comments


14/15

14


n introduction

n pre-requisites

n

governingequationsn continuityequation

n momentumequation

n summary


n Lax-Wendroff

n MacCormack

n numericalaspects


Outline


n truncationerrorcausesdissipationanddispersion

n numericaldissipationiscausedbyeven-ordertermsofthetruncationerror

n numericaldispersioniscausedbyodd-orderterms

n leadingterminthetruncationerrordominatesthebehavior

Numerical Effects

x

u

x

u

x

u

originalfunction dissipation dispersion


n artificialviscositycompromisestheaccuracy,but

improvesthestabilityofthesolution

n addingartificialviscosityincreasestheprobabilityofmakingthesolutionlessaccurate,butimprovesthestability

n similartoiterativesolutionschemesforimplicitintegrationschemes,whereasmallernumberofiterationsintroduces

artificialviscosity,butimprovesthestability

Numerical Effects


n governingequationsforcompressibleflow

n continuityequation,momentumequation

n Navier-Stokes(viscousflow)andEuler(inviscid flow)

n discussionofconservationandnon-conservationform

n explicittime-marchingtechniques

n Lax-Wendroff

n MacCormack

n numericalaspects


Summary


15/15

15


Recent Research

M.Carlson,P.J.Mucha,G.Turk,RigidFluid:Animatingthe

InterplayBetweenRigidBodiesandFluid,Siggraph 2004.


Recent Research

T.G.Goktekin,A.W.Bargteil,J.F.OBrien,

AMethodforAnimatingViscoelastic Fluids,Siggraph 2004.

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