Computer Methods in Applied Mechanics and Engineering(8)

Wenotethatcirculationisonlystrictlypreservedifthereisnovorticityadvectingordi usingoutoftheoriginaldomain.Duringvorticitytransferfrom netocoarsemesh,circulationisonlypreservedtothelevelofdiscretizationerror,sincethediscretizationerrorisdi erentoneachmeshandadvectionanddi usionratesarethereforeslightlydi erent.Testsbelowcon rmthatchangesincircu-lationasstructurespassbetweenthedi erentdomainsareappropriatelysmall.

Utilizingthemulti-domaindescriptionofthecirculationandsolutionofthePoissonequation,wenowwritetheoverallsystemofequationstobesolvedateachtime-step.S IþbDt

K

ScðkÞ

ü 2IÀbDtCTC

cðkÞnþDtð3CTNðqðkÞnÞÀCTNðqðkÞnÀ1

22

ÞÞþDt

bckÞðcðkþ1ÞÃÞ þ½Pðkþ1Þ!ðkÞðcðkþ1Þn2

cð½Pðkþ1Þ!ðÞ Þ; k¼Ng;NgÀ1;...;1;

ð35ÞECSKIþbDtK À1

!SðECÞT~f¼ECSKScð1ÞÃþ1

2

ÀunB;ð36Þcnþ1¼cð1Þ

Ã

ÀS IþbDt2

K À1

SðECÞT~

f;ð37Þsnþ1¼SKScnþ1:

ð38Þ

Notethatinsolvingforthestreamfunctionatthenexttimestep,Eq.(38),wesavethecoarsi edcirculation eldsandstreamfunctionstouseontheright-handsideofEq.(35)atthenexttime

step.

一些ME专业提升的论文。

2140T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

Whenvorticity(j)crossestheboundaryofagivengridlevel,thec eldsarenotnecessarilysmoothacrosstheinterfaces,especiallyatthecoarsestlevels.Thepropagationofavortexthroughmeshlevelsisexaminedinthenextsec-tionanditispossibletoseesomeslightinternalre ectionsofthelocalcirculationneartheboundary.However,theerrorsremaincon nedtoasmallregionneartheboundaryanddi usedovertimebythephysicalviscosity.

Themulti-domaintechniquecomeswithasigni cantincreaseincomputationalexpense.SincewenowsolvetheintermediatevorticityequationeachPoissonequationNgtimes,theoperationcountgoesupbyafactorofNg.Nevertheless,itenablesustoutilizethefastalgorithmdescribedintheprevioussection.Moreover,we ndthatthemulti-domainissu cientlyaccuratethatcomputa-tionaldomaincanbemadesnugaroundthebody.Runtimesforparticularexamplesarediscussedbelow.

Wenotethatinmanysituations,itisdesirabletospecifyauniform owaboutabody.Thisissimpletoaccomplishinthenullspaceformulation,asthereisnocirculationasso-ciatedwithit.Oneneedonlyaddtheuniform owtoqn

resultingfromEq.(27)andtounþ1

inEq.(36)onecouldaddanypotential owB

.Inprincipleinthisway,atleastpro-videditsatis esthediscretePoissonequationwithzeroright-handside.5.Validationexamples

5.1.Velocity eldforanOseenvortex

Thetwo-dimensionalvelocity eldassociatedwithaGaussiandistributionofvorticity(Oseenvortex)iscom-putedwiththemulti-domainboundaryconditions.Thistestisusedtovalidatethemethodologysinceitispossibletoderiveanalyticallytheexpectedimprovementinmulti-domainsolutionwithincreasingNgforthiscase.Asdis-cussedabove,thelargestdomainusesnopenetration/nostressboundaryconditions.Ananalyticalsolutionforthevelocity eldwiththeseboundaryconditionsmaybecon-structedbythemethodofimagessuchthattheexpectederrorforthemulti-domainboundaryconditionscanbeevaluated.TheprocedureisstraightforwardandisdescribedinAppendixB.Theresultsshowthattheerrorshoulddecreaseas4ÀNgingeneral,andforthespecialcaseofasquaredomain,therateimprovesto16ÀNg.Thevorticity eldisinitializedwithxðx;yÞ¼

C4pmeÀr2

;ð39Þwherer¼pt

x2þy2isthedistancefromtheorigin.Theanalyticalsolutionfortheazimuthalvelocityis

uC r2 hðx;yÞ¼2pr

1ÀeÀ:ð40Þ

Westartthecomputationattimet=t0andchooseCandt0suchthatthemaximumspeedisUatr=R.Inwhatfol-lows,alllengthsandvelocitiesarenormalizedbyRandU,

respectively.Thevorticityisevaluatedattheverticesofarectangulardomainwithuniform(andequal)gridspacinginbothdirectionsandthePoissonequationissolvedusingthemulti-domainmethoddiscussedabove.InFig.6,con-toursofthevelocityinthexdirectionareplottedforacasewithNg=5;thevelocitycomputedoneachofthe vedo-mainsareoverlaidtoshowthatthevelocity eldremainssmooththroughthedomaintransitions.InFig.7,theL1errorofu(theentirediscretevelocity eld)isplottedasNgisvariedfrom1to5,fortwodi erentcomputationaldomains.Fortherectangulardomainextendingto±4and±8intheNxandydirections,respectively,thedecayfol-lowsthe4ÀgtheoreticalestimatethroughNg=5.ForthesquaredomainÀNextendingto±5ineachdirection,weob-servethe16gdecaydowntoerrorsaround10À3whichcanbeshowntoberoughlythelevelofthetruncationerrorforthesecond-order nitevolumemethodatthisgridden-sity.Forthenon-squaredomain,werequireaboutNg=5toreducetheboundaryconditionerrortoasimilar

level.

Fig.6.MultidomainsolutionofthePoissonequationwithNg=5foranOseenvortex.Contoursofthevelocitycomponentinthexdirectionareplottedforeachofthe5grids.Thesmallestgridextendsto±5R,withgridspacingD=0.05R.Contourlevels:min=À0.2,max=0.2,increment=

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