Cfd Lecture

7/21/2019 Cfd Lecture

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TheUNIVERSITY of NORTH CAROLINA at CHAPEL

Intro to

Computational FluidDynamic!randon Lloyd

CO"P #$%

April &'( #))*

Image courtesy of Prof. A. Davidhazy at RIT. Used without permission.


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O+,r+i,-

Und,rtandin. t/, Na+i,r0Sto1,,2uation- D,ri+ation 34ollo-in. 56ri,7,l &%%89:

- Intuition

Sol+in. t/, Na+i,r0Sto1, ,2uation- !aic approac/,

- !oundary condition

Trac1in. t/, 4r,, ur4ac,


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Op,rator

0 .radi,nt

di+ 0 di+,r.,nc,

#0 Laplacian

; 0 Hand -a+in. < L,n.t/ymat/ compr,ion

Foundation

yu

xuuu

+== div

2

2

2

22

yu

xuu

+

=

=

yu

xuu

,


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Tranport T/,or,m

xdtxufftxdtxfdt

d

t t

),()div(),(

+

=

),(),( tct

txu

=),( tcx =

c

0

),( tc

t


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Con,r+ation o4 "a

densityis;),()0,(mass0

== t xdtxxdx

0),()div(),( = += xdtxutxdtxdtd t t

0)div( =+

u

t

Transport theorem

0div =u

Integrand vanishes

is constantfor incompressible

fluids

Continuity equation


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Con,r+ation o4

"om,ntum= t xdtxutx ),(),(momentum

= forcesactingmomentuminchange t xdtxftx ),(),(:forcesbody

t dsntx ),(:forcessurface

+= ttt dsntxxdtxftxxdtxutxdtd

),(),(),(),(),(

norma:tensorstress: n


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Con,r+ation o4

"om,ntum

0divdiv)())(()( =++ guuuuudt

d

Transport theorem Divergence

theorem

fupuudt

ud ++= 2")(

omentum equation

+= ttt dsntxxdtxftxxdtxutxdtd

),(),(),(),(),(


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Na+i,r0Sto1, E2uation

fupuudtud

u

++=

=

2")(

0

convection viscositye!terna"

forcespressure


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Sol+in. t/, ,2uation

!aic Approac/

&= Cr,at, a t,ntati+, +,locity

>,ld=a= Finit, di?,r,nc,7= S,mi0La.ran.ian m,t/od 3Stable Fluids

5Stam &%%%9:

#= Enur, t/at t/, +,locity >,ldi di+,r.,nc, 4r,,@a= Adut pr,ur, and updat, +,lociti,

7= Pro,ction m,t/od


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T,ntati+, V,locity Fi,ld

Finit, di?,r,nc, B m,c/anicaltranlation o4 ,2uation=

n


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Limit on tim, t,p

CFL conditionsB dont mo+,

mor, t/an a in.l, c,ll in on,tim, t,p

Di?uion t,rm

ytvxtu


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Sta7l, Fluid ",t/od

&= Add 4orc,@

#= Ad+,ction*= Di?uion

)()()(& 0" xftxuxu +=


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Ad+,ctionFinit, di?,r,nc, i unta7l, 4or lar., t=

Solution@ trac, +,lociti, 7ac1 in tim,=6uarant,, t/at t/, +,lociti, -ill n,+,r 7lo-up=

)),((&)(& "2 txpuxu =


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Di?uionDicr,tiin. t/, +icoity t,rm pr,ad +,locity

amon. imm,diat, n,i./7or= Unta7l, -/,n

tim, t,p too mall( .rid pacin. too lar.,( or+icoity i /i./=

Solution@ Int,ad o4 uin. an ,plicit tim, t,pu, an implicit on,=

T/i l,ad to a lar., 7ut par, lin,ar yt,m=

)(&)(&)(23

2 xuxut

=I


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Eplicitly En4orcin.

uG)Sinc, -, /a+, not y,t add,d t/, pr,ur, t,rm(

-, can u, pr,ur, to ,nur, t/at t/,+,lociti, ar, di+,r.,nc, 4r,,=

u) incr,a,d pr,ur, and u7,2u,nt outu uJ) d,cr,a,d pr,ur, and u7,2u,nt inu

R,laation al.orit/m

&= Corr,ct t/, pr,ur, in a c,ll

#= Updat, +,lociti,

*= R,p,at 4or all c,ll until ,ac/ /a uJK


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Sol+in. 4or pr,ur,

Anot/,r approac/ in+ol+, ol+in. 4or a pr,ur,corr,ction t,rm o+,r t/, -/ol, >,ld uc/ t/att/, +,lociti, -ill 7, di+,r.,nc, 4r,, and t/,nupdat, t/, +,lociti, at t/, ,nd=

)"()()"( "& +

+

=

= nnn

pt

uu

dt

ud

)"()()"(

& ++

= nnn

p

t

uu

0& )"(2)()"( =

= ++ nnn pt

uu

Discretize in time

Rearrange terms

#atisfy continuity eq.


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Sol+in. 4or pr,ur,

, ,nd up -it/ t/, Poion ,2uation 4orpr,ur,=

T/i i anot/,r par, lin,ar yt,m= T/,, typ,o4 ,2uation can 7, ol+,d uin. it,rati+,m,t/od=

U, pr,ur, to updat, >nal +,lociti,=

)()"(2 & nn ut

p

= +

)"()()"( & ++ = nnn ptuu


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Pro,ction ",t/od

T/, H,lm/olt0Hod., D,compoition T/,or,mtat, t/at any +,ctor >,ld can 7, d,compo,da@

-/,r, u i di+,r.,nc, 4r,, and qi a calar>,ld d,>n,d implicitly a@

, can d,>n, an op,rator P t/at pro,ct a+,ctor >,ld onto it di+,r.,nc, 4r,, part@

quw +=

2qw =

qwwu ==


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Pro,ction ",t/od

Applyin. P to 7ot/ id, o4 t/, mom,ntum,2uation yi,ld a in.l, ,2uation only in t,rmo4 u@

T/u 4or t/, lat t,p @

Loo1 4amiliarM T/, calar >,ld qi actuallyr,lat,d to pr,ur,

quuqu == && 2

))('(

2

fuuudt

ud

++=

)"()()"( & ++

= nnn pt

uu

)()"(2 & nn ut

p

= +


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T/, !ottom Lin,

All t/r,, m,t/od ar, ,2ui+al,nt


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!oundary Condition

No lip@ S,t +,locity to ) on t/, 7oundary=6ood 4or o7tacl,=

Fr,, lip@ S,t only t/, +,locity in t/,

dir,ction normal to t/, 7oundary to ,ro=6ood 4or ,ttin. up a plan, o4 ymm,try=

Ino-@ Sp,ci>,d poiti+, normal +,locity=6ood 4or ourc,=

Outo-@ Sp,ci>,d n,.ati+, normal

+,locity= 6ood 4or in1= P,riodic@ Copy t/, lat ro- and column o4

c,ll to >rt ro- and column= 6ood 4orimulatin. an in>nit, domain=


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Sta..,r,d 6rid

T/, ta..,r,d .rid pro+id,+,lociti, imm,diat,ly at c,ll7oundari,( i con+,ni,nt 4or>nit, di?,r,nc,( and a+oidocillation=

Conid,r pro7l,m o4 a #D uid atr,t -it/ no ,t,rnal 4orc,=T/, continuou olution i@

On a dicr,ti,d non0ta..,r,d.rid you can /a+,@

jip , jip ,"+jip ,"

", +jip

", jip

2

", +jiv

2", jiv

jiu

,2" jiu ,

2"+

constant00 === pvu

oddforeven,for

00

2",

,,

jiPjiPp

vu

ji

jiji

++=

==


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Trac1in. t/, Fr,,

Sur4ac,T/, mo+,m,nt o4 t/, 4r,, ur4ac,

i not ,plicit in t/, Na+i,r0Sto1, ,2uation=

T/r,, m,t/od 4or trac1in. t/,4r,, ur4ac,@

&= "ar1,r and c,ll 3"AC: m,t/od#= Front trac1in.

*= Particl, l,+,l ,t m,t/od


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"AC

Du, to 5Harlo- and ,lc/ &%'$9=

Trac1 mal, mar1,r particl, tod,t,rmin, -/,r, t/, 4r,, ur4ac, i

locat,d="ar1,r ar, tranport,d accordin. to t/,

+,locity >,ld=

C,ll -it/ mar1,r ar, uid cells= Fluid c,ll7ord,rin. ,mpty c,ll ar, surface cells=

T/,r, ar, 7oundary condition t/at mut 7,ati>,d at t/, ur4ac,=

Et,nd,d 7y 5C/,n ,t al= &%%9 to trac1particl, only n,ar t/, ur4ac,=


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Front Trac1in.

Propo,d 7y 5Fot,r and F,d1i- #))&9

Front trac1in. u, a com7ination o4 a l,+,l ,t andparticl, to trac1 t/, ur4ac,=

T/, particl, ar, u,d to d,>n, an implicit 4unction=

An iocontour o4 t/i 4unction r,pr,,nt t/, li2uidur4ac,=

T/, iocontour yi,ld a moot/,r ur4ac, t/anparticl, alon,=


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Front Trac1in.

Uin. t/, l,+,l ,t m,t/od( t/,iocontour can 7, ,+ol+,d dir,ctly o+,rtim, 7y uin. t/, uid +,lociti,=

Particl, and l,+,l ,t ,+olution /a+,compl,m,ntary tr,n.t/ and-,a1n,,

-L,+,l ,t ,+olution u?,r +olum, lo

- Particl, can cau, +iual arti4act

- L,+,l ,t ar, al-ay moot/=

- Particl, r,tain d,tail=


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Front Trac1in.

Com7in, t/, t-o t,c/ni2u, 7y .i+in. particl,mor, -,i./t in ar,a o4 /i./ cur+atur,=Particl, ,capin. t/, l,+,l ,t ar, r,nd,r,ddir,ctly a pla/in. dropl,t=
http://www.cs.unc.edu/~blloyd/comp259/project/foster_liquids1.avi


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Particl, L,+,l S,t

",t/odPr,,nt,d 7y 5Enri./t ,t

al #))#9=

Implicit ur4ac, lo,d,tail on coar, .rid=

Particl, 1,,p t/, ur4ac,4rom croin. t/,m 7utcant 1,,p it 4romdri4tin. a-ay=

Add particl, to 7ot/ id,o4 t/, implicit ur4ac,=

Ecap,d particl, indicat,t/, location o4 ,rror int/, implicit ur4ac, oit can 7, r,7uilt=


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Particl, L,+,l S,t

",t/odEtrapolat,d +,lociti, at t/, ur4ac,

.i+, mor, r,alitic motion=
http://www.cs.unc.edu/~blloyd/comp259/project/splash-640.avihttp://www.cs.unc.edu/~blloyd/comp259/project/glass00.avi


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R,4,r,nc,

CHEN( =( AND LO!O( N= &%%Q=Toward interactive-rate simulation of uids withmoving obstacles using the navier-stokes equations. Computer Graphics andImage Processing, 10711!.

CHEN( S=( OHNSON( D=( RAAD( P= AND FADDA( D= &%%. The surface markerand micro cell method. "nternational #ournal of $umerical %ethods in &luids,

'(, 7)*-77+.

FOSTER( N=( AND "ETAAS( D= &%%'=ealistic animation of liquids. GraphicalModels and Image Processing, )71)+.

FOSTER( N=( AND FEDI( R= #))&=ractical animation of liquids. "nProceedings of SIGGRAPH 2001, '0.

6RIE!EL( "=( DORNSEIFER( T=( AND NEUNHOEFFER( T=&%%8. NumericalSimulation in luid !"namics# A Practical Introduction. /"% %onograhs on%athematical %odeling and 2omutation. /"%

ASS( "=( AND "ILLER( 6= &%%)=aid, stable uid d3namics for comutergrahics. "n Computer Graphics $Proceedings of SIGGRAPH %0&, vol. '), )*(7.

O!RIEN( =( AND HOD6INS( = &%%$. 43namic simulation of slashing uids. "nProceedings of Computer Animation %', 1*+'0(.

STA"( = &%%%=/table uids. "n roceedings of /"556 **, 1'1-1'+.

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