Computer Methods in Applied Mechanics and Engineering(4)

Inthecaseofexternal ow,non-uniformgridstretchingisutilizedtopositionthecomputationalboundaryoDasfaraspossiblefromtheimmersedbodytominimizethein uenceofthearti cialboundaryconditionsontheinner ow eld.Allboundaryconditionsaresettouniform ow(U1,0,0)inthestreamwisedirection(x-direction)exceptfortheout owboundarywhereaconvectiveboundarycondition[33],

Ayj

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T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462135

Fig.3.TopviewofvorticalstructurebehindarectangularplateofAR=2anda=30°representedbyanisosurfaceofQ=1forRe=100atdi erenttimes.Streamlinesareoverlaidwithcolorcontourindicatingthelocalvelocitynormfrombluetoredinincreasingmagnitude.Flowdirectionfromtoplefttobottomright.(Forinterpretationofthereferencestocolourinthis gurelegend,thereaderisreferredtotheconversionofthisarticle.)

andtipvortices.TheisosurfaceherearegeneratedforunitQ-value(secondinvariantofthevelocitygradienttensor)toshow owregionswithsigni cantrotation.3Streamlinesarealsodepictedtoillustratethetip-e ects.Initiallyastrongtrailing-edgevortexisformedconvectingdown-streamwhiletheleading-edgeandtipvorticesstaystablyattachedtotheplate(t=1.5).Lateratsteady-state(t=13),thedi usedleading-edgevorticalstructureisstillstablyattachedtotheplate.Inthecaseofthree-dimen-sional ow,theviscousdi usionofvorticityinthespan-wisedirectionandthetip-e ectstabilizesthewakestructureatthislowRe.Resultswithvariousaspectratio,anglesofattack,andplanformgeometriesareexaminedinfurtherdetailin[37].

3.Nullspacemethodfortheimmersedboundarymethod3.1.Nullspaceapproach

Thenullspaceordiscretestreamfunctionapproach[11,4]isamethodforsolvingthesystem(7)withouttheimmersedboundaryformulation.Inthiscase,the owonlyneedstosatisfytheincompressibilityconstraint,whichleadsustotheuseofdiscretestreamfunction,s,suchthatq¼Cs;

ð13Þ

whichautomaticallyenforcesincompressibilityatalltime;Dqn+1=DCsn+1=0.Thisdiscreterelationisconsistentwiththecontinuousversionofthevectoridentity:$Æ$· 0.4

Pre-multiplyingthemomentumequationwithCT,thepressuregradienttermcanalsoberemovedfromthefor-mulationsinceCTGp=À(DC)Tp=0,resultinginonlyasingleequationtobesolvedforeachtimestep:CTACsnþ1¼CTðrn1þbc1Þ:

ð15Þ

Inthismethod,themostcomputationallyexpensivecom-ponentofthefractionalstepmethod,namelythepressurePoissonsolver,iseliminatedwhilethecontinuityequationisexactlysatis ed.Moreoverthefractionalsteperroraris-ingfromusinganapproximateAÀ1isnotpresentsinceanapproximateLUdecompositionisnotrequired.Thisfea-tureledChangetal.[4]tocallthistechniquetheexactfrac-tionalstepmethod.

WenotethattheoperatorCTisanotherdiscretecurloperation,andthat:c¼CTq;

ð16Þ

isasecond-orderaccurateapproximationtothecirculationineachdualcell(vorticitymultipliedbythecellareanor-maltothevorticitycomponent).

Thismethodmayingeneralbeusedonunstructuredmeshesintwoandthreedimensions[4],including,asaspe-cialcase,thesimpleCartesianmeshusedinIBmethods.Intwodimensions,thediscretestreamfunctionandcircula-tionhaveasinglecomponent(inthedirectionnormaltotheplane),whichisnaturallyde nedatthecellvertices(seeFig.4)[4].Inthreedimensionstherearethreecompo-nentsofthestreamfunctionandcirculationthatarede nedatthecentersoftheedgesoftheVoronoi(dual)cell,anal-ogouslytothevelocitycomponentsontheprimal

mesh.

Notethatwehavesetbc2=0whichisthecasefortheboundaryconditionsweconsiderhere.Moregeneralsituationsthatrequirebc250canbehandledby ndingaparticularsolutionfortheinhomogeneousvectorandaddingthesolutiontoEq.(13).

4

whereCrepresentsthediscretecurloperator.Thisopera-torisconstructedwithcolumnvectorscorrespondingtothebasisofthenullspaceofD.Changetal.[4]shouldbeconsultedfordetails.Hence,theseoperatorsenjoythefol-lowingrelation:DC 0;

3

ð14Þ

TheQ-value(thesecondinvariantof$u)isde nedas

22

Q 1ðkXkÀkSkÞ,forincompressible owwhereXandSaretheasymmetricandsymmetriccomponentsof$u,respectively[13].Comparedtothevorticitynorm,positiveQ-valuescanhighlightvorticalstructuresbyremovingregionsofhighshear.

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Fig.4.Locationofvariablesonstaggered3Dmesh.Velocitycomponentsarede nedatthecenterofeachedge.Streamfunctionandcirculationarede nedsimilarlyfortheVoronoicell–inthiscaseacellthatiso setbyhalfacelllengthineachdirection.

3.2.NullspaceapproachwithanimmersedboundaryInordertosatisfyboththeincompressibilityandtheno-slipconditionswiththenullspacetechnique,itwouldbenecessarytoderiveabasisforthenullspaceofQT.Although,asingularvaluedecompositionofQTcanbeperformedtonumericallydeterminethenullspace,theresultisnotingeneralasparserepresentationwhichisdesirableforcomputationalfeasibility.Ananalyticalderi-vationofthenullspaceoperatordoesnotseemtobeaneasytaskeither.Moreover,inthegeneralcasewherethebodyismoving,thenullspacerepresentationwouldneedtoberecomputedatleastoncepertimestep.

Tocircumventthisdi culty,weonceagainrelyonaprojectionapproach.TConsiderthesystemn+1thatisobtainedbyincorporatingCandqn+1=CstoEq.(8).TheincompressibilityconstraintandthepressurevariableareeliminatedandwearriveatanotherKTTsystem:

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