Computer Methods in Applied Mechanics and Engineering

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一些ME专业提升的论文。

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Available online at

Comput.MethodsAppl.Mech.Engrg.197(2008)

2131–2146

/locate/cma

Afastimmersedboundarymethodusinganullspaceapproach

andmulti-domainfar- eldboundaryconditions

TimColonius*,KunihikoTaira

DivisionofEngineeringandAppliedScience,CaliforniaInstituteofTechnology,CA91125,USAReceived21March2007;receivedinrevisedform3August2007;accepted6August2007

Availableonline12September2007

Abstract

Wereportonthecontinueddevelopmentofaprojectionapproachforimplementingtheimmersedboundarymethodforincompress-ible owsintwoandthreedimensions.BoundaryforcesandpressureareregardedasLagrangemultipliersthatenabletheno-slipanddivergence-freeconstraintstobeimplicitlydeterminedtoarbitraryprecisionwithnoassociatedtime-steprestrictions.Inordertoaccel-eratethemethod,wefurtherimplementanullspace(discretestreamfunction)methodthatallowsthedivergence-freeconstrainttobeautomaticallysatis edtomachineroundo .Byemployingafastsinetransformtechnique,thelinearsystemtodeterminetheforcescanbesolvede cientlywithdirectoriterativetechniques.Amulti-domaintechniqueisdevelopedinordertoimprovefar- eldboundaryconditionsthatarecompatiblewiththefastsinetransformandaccountfortheextensivepotential owinducedbythebodyaswellasvorticitythatadvects/di usestolargedistancefromthebody.Themulti-domainandfasttechniquesarevalidatedbycomparingtotheexactsolutionsforthepotential owinducedbystationaryandpropagatingOseenvorticesandbyanimpulsively-startedcircularcyl-inder.Speed-upsofmorethananorder-of-magnitudeareachievedwiththenewmethod.Ó2007ElsevierB.V.Allrightsreserved.

1991MSC:76D05;76M12PACS:47.11.+j

Keywords:Immersedboundarymethod;Fractionalstepmethod;Projectionmethod;Nullspacemethod;Vorticity/streamfunctionformulation;Far- eldboundaryconditions;Multi-domainmethod;FastPoissonsolver;Finitevolumemethod;Incompressibleviscous ow

1.Introduction

Intheimmersedboundarymethod(IBmethod),immersedsurfacesaregeneratedbyforcesatasetofLagrangianpoints[29,20,19].The owissolvedonanEuleriangridthatdoesnotconformtothebodygeometry–typicallyauniformCartesiangridisused.Theboundaryforcesthatexistassingularfunctionsalongthesurfaceinthecontinuousequationsaredescribedbydiscretedeltafunctionsthatsmear(regularize)theforcinge ectovertheneighboringEuleriancells.

Correspondingauthor.

E-mailaddresses:colonius@caltech.edu(T.Colonius),kunihiko@cal-tech.edu(K.Taira).

0045-7825/$-seefrontmatterÓ2007ElsevierB.V.Allrightsreserved.doi:10.1016/j.cma.2007.08.014

*

IntheoriginalIBmethod,surfaceswereviewedas ex-ibleelasticmembraneswithaconstitutiverelation(e.g.Hooke’slaw)relatingtheforcestothemotionoftheLagrangianpoints[28].Thistechniquewaslaterextendedtosurfaceswithprescribedmotion(andinparticularrigidbodies)bytakingthespringconstanttobelarge[2,16].Goldsteinetal.[9]appliedtheconceptoffeedbackcontroltocomputetheforceontherigidimmersedsurface.Thedi erencebetweenthevelocitysolutionandtheboundaryvelocityisusedinaproportional-integralcontroller.Con-stitutivelawsareeliminatedinthedirectforcingmethod[21,7];forcinginthemomentumequationisdeterminedbypenalizingtheslipatthe(interpolated)surface.Fortheaforementionedtechniquesthatutilizeconstitutiverela-tions,thechoiceofgain(sti ness)rgegain

一些ME专业提升的论文。

2132T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

resultsinrestrictionsonthetimestep,whilesmallgainresultsinsliperror.1Directforcingmethodssimilarlyresultinasliperroratthesurface.Whilethesliperrorisreportedtobesmall[7],themagnitudecannotbeestimatedinadeductivemanner.FurtherinformationregardingtheIBmethodandhigher-orderextensionsaregiveninarecentreview[20].

AnalternativeistoregardtheboundaryforcesasLagrangemultiplierswhosevaluesarechosentosatisfytheno-slipconstraint[8,36].Byintroducingappropriateregularizationandinterpolationoperatorsandgroupingthepressureandforceunknownstogether,thediscretizedincompressibleNavier–Stokesequationscanbeformulatedwithastructurealgebraicallyidenticaltothetraditionalfractionalstepmethod[36].Thepressureandforceunknownsarefoundbysolvinga(modi ed)Poissonequa-tion.Inwhatfollows,werefertothismethodastheimmersedboundaryprojectionmethod(IBPM).

TheprincipleadvantagesoftheIBPMtechniquearethatthecontinuityandno-slipconstraintscanbesatis ed(toarbitraryaccuracy)implicitlyatthenexttimelevel,andthattheCourantnumberisonlylimitedbythechoiceoftimemarchingschemesfortheviscousandadvectiontermsinthemomentumequation.Further,itispossibletoarrangealloperationssothatthemethodisuniformlysec-ond-orderaccurateintime,andsothatthematrixarisingfromimplicittreatmentoftheviscoustermsinthemomen-tumequationaswellasthemodi edPoissonmatrixarebothsymmetricandpositivede nite.Consequentlytheconjugate-gradientmethodcanbeusedtosolvethelinearsystems.However,iterativesolutionofthelinearsystemsresultsinaconvergenceerror.Thispresentsnodi cultyinthemomentumequationwherethesolutionneedonlybeconvergedtotheextentthatitissmallerthanotherdis-cretizationerrors.Butinthemodi edPoissonequation,convergenceerrorsdirectlyimpacttheaccuracytowithwhichthedivergence-freeandno-slipconstraintsaresatis- ed.Whiletheerrorscanbemadearbitrarilysmall,largenumbersofiterationsmayberequired.

Inthepresentpaper,werevisitthismethodandproposesomeimprovementstoacceleratetheIBPM.InSection2,wereviewtheoriginalformulationandpresentsomenewresultsfromarecentextensionofthemethodtothree-dimensional ows.InSection3,weimplementanullspace(discretestreamfunction)method[11,4]thatallowsthedivergence-freeconstrainttobeautomaticallysatis edtomachineroundo .Weshowthatifthegridiskeptuniformthroughoutspace(withequalspacinginalldirections),thePoisson-likeequationfortheforcescanbee cientlysolvedeitherdirectlyforstationarybodiesoriterativelyformov-ingbodiesthroughtheuseofafastsinetransform.Whileuniformgridspacingisinfactrequiredinthevicinityof

1

Sti nessissuesarealsoobservedwithelasticsurfaces.Recently,stablesemi-andfully-implicittemporaldiscretizationstocouplethevelocity eldandtheboundaryforceforelasticboundarieshavebeenproposedby[24,22].

thebodybythediscretedeltafunctionthatisusedtoreg-ularizethesurfaceforce,itisrelativelyine cientforexter-nal owswherethedomainneedstoextendtolargedistancefromthebody.IntheoriginalIBPM,thisdi -cultyisovercomebystretchingthemeshawayfromthebody,butthisisincompatiblewiththenullspace/fastsinetransformformulationintroducedhere.Toovercomethisrestriction,wederiveinSection4improvedfar- eldboundaryconditionsthatarecompatiblewiththefastmethodandallowthedomaintobemoresnugaroundthebody.Thenewboundaryconditionsaccountfortheextensivepotential owinducedbythebodyaswellasvor-ticitythatadvects/di usestolargedistancefromthebody.Theboundaryconditionsrelyonamulti-domainapproachwherebythePoissonequationissolved(withthefastsinetransform)onaseriesofincreasinglylarger,butcoarser,computationaldomains.ValidationexamplespresentedinSections5and6demonstratethee cacyandimprovede ciency,respectively,oftherevisedformulation.2.Immersedboundaryprojectionmethod2.1.Projectionapproach

WeconsidertheincompressibleNavier–Stokesequa-tionswithasingularboundaryforcefaddedtothemomentumequationasacontinuousanalogoftheimmersedboundaryformulation:

ouotþuÁru¼Àrpþ12

Z

Reruþfðnðs;tÞÞdðnÀxÞds;ð1ÞsrÁu¼0;uðnðs;tÞÞ¼

Z

ð2Þ

uðxÞdðxÀnÞdx¼uBðnðs;tÞÞ;ð3Þx

whereuandparethevelocityandpressurevariables,

respectively.Notethatweexpresstheno-slipconditionusingadeltafunctionconvolutionalongtheimmersedsur-face.Here,non-dimensionalizationisperformedtoyieldasingleparameterofReynoldsnumber,Re.Spatialvariablexrepresentspositioninthe ow eld,D,andndenotescoordinatesalongtheimmersedboundary,oBhavingavelocityofuB.ThegeometryoftheimmersedobjectBisconsideredtobeofarbitraryshape.Inthepresentdevelop-ment,therearenoforcesinteriortothebodyandanymo-tionordeformation2ofthebodyisprescribed.Furthergeneralizationsofthemethodarepossiblebutawaitfuturework.

Theabovesystemisdiscretizedwithastandardstag-geredCartesiangrid nitevolumemethod.ThemeshandvariablelocationsaredepictedinFig.1.Thecomputa-tionaldomain,D,isrepresentedbyaCartesiangrid,(xi,yi),andtheimmersedboundary,oBisdescribedbyasetofLagrangianpoints,(nk,gk),whichcanbeafunctionof

2

Forexamplefullycoupled uid–structureinteractionviaanimmersedcontinuummethod[38].

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T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462133

time.ThebodyBisassumedtohaveaprescribedsurfacemotion.Followingthematrix–vectornotationof[4],wecanwriteEqs.(1)–(3)semi-discretelyasM

dq

dt

þGpÀHf¼NðqÞþLqþbc1;ð4ÞDq¼0þbc2;

ð5ÞEq¼unþ1

B;

ð6Þ

whereq,p,andfarethediscretevelocity uxvector,pres-sure,andboundaryforce.Thediscretevelocity,u,canberelatedtoqbymultiplyingthecellfaceareanormaltothevector,i.e.,q=(qu,i,qv,i)=(uiDyi,viDxi).Theabove rst,second,andthirdequationsrepresentthediscretizedmomentumequation,continuityequation,andno-slipcon-ditionalongoB.Discretizednon-linearconvectivetermÀuÆ$uisdenotedbyNðqÞandoperatorsMandLarethe(diagonal)massmatrixanddiscreteLaplacian,respectively.

Wenotethatallofthematricesintheabove(andallthatfollow)aresparseandaremoste cientlycodedaspoint-operators-subroutinesreturnthematrix–vectormul-tiplysuchthatthematricesareneverexplicitlyformed.Forconvenience,point-operatorrepresentations(forthecaseofauniformgrid)aregiveninAppendixA.

OperatorsGandDarethediscretegradientanddiver-genceoperatorsTandcanbeformulatedsuchthatG=ÀD[27,4].TheremainingoperatorsofEandHaretheinterpolationandregularizationoperatorsresultingfromtheregularizationoftheDiracdeltafunctionsinEqs.(1)and(3).Theno-slipconstraintisenforcedbyequatingtheboundaryvelocity,uB,tothevelocityvaluealongoBinterpolatedbyEfromtheneighboringcells.Ontheotherhand,theregularizationoperatorsmearsthee ectofthesingularboundaryforcealongoBtotheCartesiangrid.Topreservesymmetryinthe nalalgo-

rithm,weconstructtheseoperatorstosatisfyE=ÀHT;see[36]forfurtherdiscussion.WementionthatmatricesG,D,E,andHarenotsquare.Consequently,Eqs.(4)–(6)2canbewritten30asasystemofalgebraicequations:nþ110n16AGÀHr0bc11

4D0

07Bq

CBCBC

E005@pfA¼@0unAþ@bc2A:ð7ÞB

þ10SubmatrixA¼1

mentofthevelocityMÀaLLresultsfromtheimplicittreat-term.Hereweapplytheimplicittrap-ezoidruleontheviscoustermwithaL¼Thetermisdiscretizedwiththesecond-order2

convectiveAdam–Bashforth(AB2)r¼Âmethod.ÃInthiscasetheright-handsidevectorn11qn

DtMþ2Lþ3NðqnÞÀ1NðqnÀ1Þ.TheAB2meth-odisnotself-starting2andwereplace2

itwithbackwardEu-lerforthe rsttimestep.Theinhomogeneoustermsbc1andbc2dependontheparticularboundaryconditionsandarediscussedin[36].Boundaryconditionsaredis-cussedingreaterdetailinSections3and4.

WiththeuseofstaggeredCartesiangrid,weareabletogloballyconservemass,momentum,kineticenergy,andcirculation[17,23,26].Detaileddiscussiononspatialdis-cretizationsofvariousformsofthenon-linearconvectiveterm(rotational,divergence,skew-symmetric,andadvec-tiveforms)areprovidedin[23,26].Theexplicitright-handsidetermingeneralalsoincludesinhomogeneousterms,bc1andbc2,generatedbytheboundaryconditionsfromthediscreteLaplacianLandthedivergenceDoperators,respectively.

Byapplyingthepropertiesofthesub-matrices,Eq.(7)can2berestated3as

0nþ1106AGET

qrn

þbc14G

T0075B

@pCA¼B1@Àbc2CAð8ÞE00~funB

þ1;where~f

istheboundaryforcewithanincorporatedscalingfactor.ThisformoftheequationisknownKahn–Tucker(KKT)systemwhereðp;~astheKarush–

f

ÞTappearasasetofLagrangemultipliertosatisfyasetofkinematiccon-straints.Inthediscretizedsetofequations,theconstraintsarepurelynumericalanditisnolongernecessarytodistin-guishthepressureandboundaryforce. neacombinedvariablek¼ðp;~Insteadwecande-f

ÞTfortheLagrangemultipliersandgroupthesubmatricesasQ=[G,ET].Notethatbyremovingtheboundaryforceandno-slipconditionalongoB,thetraditionaldiscretizationoftheincompress-ibleNavier–Stokesequationscanberetrieved.

SincewenowhaveformulatedtheimmersedboundaryformulationoftheNavier–Stokesequationsinanalgebra-icallyidenticalmannertothetraditionaldiscretizationoftheincompressibleNavier–Stokesequations,standardsolutiontechniquescanbeutilized.Hereweapplythepro-jection(fractional-step)algorithmtoEq.(8),whichcanbeexpressedasanapproximateLUdecompositionoftheleft-handsidematrix[27],toproducetheimmersedboundaryprojectionmethod[36]

:

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Aqür1;ðSolveforintermediatevelocityÞQ

T

ð9Þð10Þ

AyjQk

¼QqÀr2;ðSolveamodifiedPoissonequationÞ

ð11Þ

TÃ

ouou

þU1¼0;otox

ð12Þ

qnþ1¼qÃÀAyjQk;ðProjectionstepÞ

hasbeenapplied.Boundaryconditionsalongthecomputa-tionalboundaryarediscussedingreaterdetailinSection4inthecontextofthenewformulation.2.2.Three-dimensionalIBPM

Two-dimensionalvalidationexamplesandconvergencestudiesfortheIBPMarepresentedin[36].TodemonstratethattheIBPMcanbeimplementedinthreedimensions,webrie ydescriberesultsforthree-dimensional owoveralow-aspect-ratio atplateatangleofattack.Asanexam-ple,arectangular atplateofaspectratio,AR=2,atanangleofattackofa=30°isinstantaneouslygeneratedinauniform ow eldatt=0.TheReynoldsnumberissettoRe=100andthecomputationaldomainistakentobe[À4,6.1]·[À5,5]·[À5,5](normalizedbythechord)withagridsizeof125·55·80(streamwise,vertical,andspan-wisedirections,respectively).Here,gridstretchingisappliedtoregionsawayfromtheplate,whilekeepinguni-formresolutioninthecloseproximityoftheimmersedbody.Thetimestepandtheminimumgridsizearesetto0.01and0.04,respectively,tolimitthemaximumCourantnumberto0.5duringthesimulation.

InFig.2,thespanwisevorticitycontoursatthemidspanarecomparedtodigitalparticleimagevelocimetry(DPIV)measurementsacquiredfromacompanionexperimentper-formedinanoiltowtank.SimulationresultsandtheDPIVdataarefoundtobeinagreementalongwithforcemea-surementsontheplatevalidatingthethree-dimensionalimmersedboundaryprojectionmethod.Thecorrespondingthree-dimensionalwakestructuresarepresentedinFig.3toillustratetheformationofleading-edge,

trailing-edge,

denotesthejthorderTaylorseriesexpansionofwhere

À1

AwithrespecttoDt.Theexplicittermsontheright-handsidehavebeengroupedintor1andr2.In[36],AandQTAyjQareconstructedtobesymmetricpositivede niteoperatorsinordertousetheconjugate-gradientmethodtoe cientlysolvefortheintermediatevelocityandtheLagrangemulti-pliers.Incontrasttothetraditionalimmersedboundarymethods,heretheno-slipconditionalongoBisenforcedonthesolutionbyprojectingtheintermediatevelocity eldintothesolutionspacethatsatis esbothdivergence-freeandno-slipconstraints.

TheIBPMisfoundtobesecond-orderaccurateintimeandbetterthan rstorderaccurateinspaceintheL2mea-sure.Sincethereisnoneedforanyconstitutiverelations(e.g.,Hooke’slaw[2,16]andproportional-integralcontrol-ler[9])tocomputetheboundaryforce,sti nessissuesarecircumventedallowingtheCourantnumbertobelimitedonlybythechoiceoftimemarchingschemesforthevis-cousandconvectiveterms.

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

Ayj

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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:

"CTACCTE

T#

snþ1 CTrn!1EC0~f¼unBþ1:ð17ÞTheleft-handsidematrixissymmetricbutingeneralindef-inite,makingadirectsolutionlesse cient.Theprojection

(fractionalstep)approachmimicsEqs.(9)–(11),andweobtain

CTACsüCTrn1;

ð18ÞECðCTACÞÀ1ðECÞT~f¼ECsÃÀunþ1B

;ð19Þsnþ1¼sÃÀðCTACÞÀ1ðECÞT~f

;ð20Þ

wherewehaveasnotyetinsertedanapproximationfortheinverseofCTAC.Directsolutionofthissysteminthegen-eralcaserequiresanestediterationtosolvethemodi edPoissonequation.Thismaybefeasibleingeneral(aroughoperationcountindicatethattheworkissimilartoEqs.(9)–(11)).Inthecasewherethebodyisnotmoving,itismoreoverpossibletoperformaCholeskydecompositionofEC(CTAC)À1(EC)Tonceandforall,sincethedimensionofthesystemscaleswiththenumberofforcesfortheim-mersedboundary.Inthiscaseasystemofequationsof

theformCTACx=bneedbesolvedonceforeachLagrangianforceatthebeginningofthecomputation.3.3.Fastmethodforuniformgridandsimpleboundaryconditions

Inthissectionwereverttothesemi-discretemomentumequation,

M

dqþGpþETdt

~f¼NðqÞþLqþbc1;ð21Þ

wheresymbolsareasde nedpreviously.Thedivergence

freeandno-slipconstraintsareunchanged.

Wenowshowthatwithsimpli cation,asimilarsystemtoEqs.(9)–(11)maybesolvedusingfastsinetransforms,resultinginasigni cantreductionincomputationalwork.Whenthegridisuniform(withequalgridspacinginallcoordinatedirections),themassmatrixMistheidentitymatrix.Weassumeforthemomentthatthevaluesofthevelocityareknownintheregionoutsidethecomputationaldomain.WeapplysimpleDirichletboundaryconditionstothevelocitynormaltothesides/edgesofthecomputationaldomain,ckingfurtherinformation,onecouldspecify,forexample,ano-penetrationBCforthenormalcomponentofvelocityandazerovorticity(orno-stress)conditionfortheremainingtangentcomponents.Thesearenaturalboundaryconditionsforanexternal owaroundthebody,providedthedomainislarge.Inthenextsectionwewillshowhowimprovedestimatesforthevelo-citiesoutsidethecomputationaldomaincanbeobtainedviaamulti-domainapproach.

Withthesesimpli cationsweoperateonEq.(21)withCT

(whicheliminatesthepressure)andweobtaindcþCTdt

ET~f¼ÀbCTCcþCTNðqÞþbcc:ð22Þ

InderivingthisequationwehaveusedthatLq=ÀbCCTq=ÀbCcprovidedthatDq=0.Herebisacon-stantequalto1/(ReD2),whereDistheuniformgridspac-ing.2Thisidentitymimicsthecontinuousidentity$u=$($Æu)À$·$·u=À$·$·u.

Withuniformgridandtheaforementionedboundaryconditions,thematrixÀbCTCisthestandarddiscreteLaplacianoperatorona5-or7-pointstencilintwoandthreespatialdimensions,respectively.Theboundarycondi-tionsdiscussedaboveresultinzeroDirichletboundaryconditionsforc.ThisdiscreteLaplacianisdiagonalizedbyasinetransformthatcanbecomputedinOðNlogc)[30].Wedenote2NÞoperations(whereNisthedimensionofherethesinetransformpair:^c¼Sc$c¼S^c;

ð23Þ

wherethecircum exdenotestheFouriercoe cients.Inwritingthetransformpair,wehaveusedthefactthatthesinetransformcanbenormalizedsothatitisidenticaltoitsinverse.Further,wemaywritesymbolicallyK=

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462137

SCTCS,whereKisadiagonalmatrixwiththeeigenvaluesofCTC.Thesearepositiveandknownanalytically(e.g.[30]),andwenotethatthereisnozeroeigenvalue(sincetheboundaryconditionsareDirichlet).

Applyingthesametime-marchingschemesusedprevi-ouslyS weobtainthetransformedsystem:IþbDtK Scü IÀbDt

CTC

cn22

þDtÀ2

3CT

NðqnÞÀCTNðqnÀ1ÞÁ þDtbcc;

ð24ÞECSKÀ1 IþbDt À1

!2

KSðECÞT~f¼ECSKÀ1ScÃÀunþ1B;ð25Þcnþ1

¼cÃÀS IþbDt2

K À1

SðECÞT~

f:ð26Þ

Thevelocity,neededforthenexttimestep,maybefoundbyintroducingthediscretestreamfunction:qn¼Csnþbcq;

sn¼SKÀ1Scnþbcs:

ð27Þ

Eachofthevectorsbcc,s,qinvolvestheassumedknownval-uesofvelocityattheedgeofthecomputationaldomain.Theirvaluesarediscussedindetailinthenextsection.Inthenewsystemofequations,onlyonelinearsystemneedbesolved,Eq.(25),withapositivede niteleft-handsideoperator.Thatthematrixispositivede nitecanbeseenbyinspection.ThedimensionsofthematrixarenowNf·Nf,andthusmanyfeweriterationsarerequiredthantheoriginalmodi edPoissonequation,Eq.(10).Tobemoreprecise,eachiterationonEq.(25)requiresOðNð2log2NþNbwþ4dÞÞoperations,whereNisthenum-berofvorticityunknownsandNbwisthebandwidth5ofthebody-forceregularization/interpolationoperators,anddisthedimensionalityofthe ow(2or3for2Dor3D,respectively).ForthediscreteDeltafunctionwithasup-portof3D,wehaveNbw=3d.FortheoriginalPoissonequation,Eq.(10),thecostperiterationisOðNÂðNbwþð2dþ1ÞjÞþ4dÞ,wherejistheorderoftheapproximateTaylor-seriesinverseofAandthefactor2d+1isthestencilofthediscreteLaplacian.Furthermore,usingstandardestimatesforthenumberofiterationsrequiredforconvergenceoftheconjugate-gradientTmethod[35]alongwiththeknowneigenvaluesofCC,wecanesti-matethattheoperationcountpertimestepforthePoissonsolutionhasbeenreducedfrom6

OðN1=2Nð7dþð2dþ1ÞjÞÞoperationcountforEq:ð10Þto

5

Wehaveusedthefactthat6HerethefactorsN1/2orN1=N2

f(Ninarrivingattheestimate.

faretheestimatednumberofiterationsoftheconjugategradientsolver,the2Nlog2Nfactorcomesfromtwo(fast)sinetransforms,the(2d+1)jfactorfromtheLaplacian,and7dfromtheinterpolation,regularization,CT,andCoperationstogether.

OðN1=2

fNð2log2Nþ7dÞÞ

operationcountforEq:ð25Þ:

Forexample,inathree-dimensionalcasewithN=1283,Nf=103,d=3,andj=3theestimatedspeedupisabout30.Foratwo-dimensionalcasewithN=1282,Nf=200,d=2andj=3,thespeedupisabout10.ThisisforthePoissonsolvealone.AdditionalspeedupoccursbecauseitisnolongernecessarytosolveasystemAx=bforthemomentumequation.NumericalexperimentsinSection6forthetwo-dimensionalcasecon rmatleasttheorder-of-magnitudeofthespeedup(theactualspeedupisfasterthanpredicted).Finally,werecallthatthenewsystemofequationsresultsinnoiterativeerrorinsatisfyingthedivergence-freeconstraint(itisautomaticallyzerotoround-o ).

Ifthebodyisstationary,thenthePoisson-likeequationfortheforcescanbee cientlysolvedusingatriangularCholeskydecomposition.Thisresultsinavastlylowerworkpertime-step,sincetheoperationcountforthePois-sonsolveissimplyOðN2fÞ.InthiscasethecomputationalspeedislimitedonlybythesolutionofEq.(24).

Tosummarize,ifthegridisuniformandsimplebound-aryconditionsareused,itisvastlypreferabletosolveEqs.(24)–(26).Werefertothisinwhatfollowsasthefastmethod.Unfortunately,forexternal ows,thesimpli edboundaryconditionsarenote ectiveunlessthecomputa-tionaldomainisquitelarge.Sincethegridisalsorequiredtobeuniform,evenfarawayfromthebody,thelargerdomainwouldquicklynegatethebene toffastmethod.However,inthenextsectionwediscussanalternativestrat-egyforimplementingboundaryconditionsinthefastmethodthathasamoremodestcostpenalty.

4.Far- eldboundaryconditions:amulti-domainapproachThefastmethodreliesonsimpli edfar- eldboundaryconditions,namelyknownvelocitynormaltotheboundaryandknownvorticity.Thesecanbesettozeroifthecompu-tationaldomainissu cientlylarge.Forsmallerdomains,thiswillleadtosigni canterrorsand,inparticular,theforcescomputedonthebodywillsu erasigni cantblock-ageerror.Theerrorarisesfromtwosources.The rstistheextensive,algebraicallydecayingpotential owinducedbythebody(orequivalently,thesystemofforces).Thesecondisthatvorticitymayadvectordi usethroughthebound-ary.InouroriginalmethoddiscussedinSection2,theseerrorsareminimizedbyusingalargedomainwithahighlystretchedCartesianmeshnearthefar- eldboundaries(butretaininguniformgridspacingnearthebody),aswellasbyusinganapproximateconvectiveout owboundarycondi-tion.Unfortunately,stretchedmeshesareincompatible7withdirectFouriermethodsforsolutionofthePoissonequation.Inthissection,weshowhowtoposeanaccurate

7

IncertainspecialcircumstancesstretchedmeshescanbecombinedwithFourier-transformmethodsforellipticequations,e.g.[3].

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far- eldboundaryconditionthatisalsocompatiblewiththefastmethoddescribedinthelastsection.

Westartbybrie yreviewingrelevantboundarycondi-tionsdesignedtoreduceoneorbothoftheaforementionederrors.Forerrorsassociatedwiththeslowlydecayingpotential ow,afewtechniqueshavebeenposedinthepasttopatchinthepotential owextendingfromthetruncatedcomputationalboundarytoin nity.RennichandLele[31]proposeatechniquefortwounboundeddirectionsandoneperiodicdirection.Theirmethodisbasedonmatchingthenumericalsolutiontoanalyticalrepresentationofthesolu-tiontoLaplaceequationoutsideacylindricalvolume.Theyreporta50%increasepertimestepforatypicallarge-scalecomputation,butthiscostismorethano setbytheabilitytousemuchmorecompactdomains.Wang[39]presentsasimilarapproachfortwo-dimensional owintheformofacorrectiontoatrialsolutionthatsatis esanincorrectDirichletboundarycondition.Vortexparticlemethodsinprincipleautomaticallyaccountfortheextensivepotential owgeneratedbythevorticity.However,inpracticeitisoftennecessarytoremoveparticlesthatadvecttolargedis-tancefromtheregionofinterest.Aninterestingtechniquetoreduceerrorsassociatedwithremovalofparticlesiscalledmerging,wherebythecirculationsofseveralvortexparticlesarecombinedintoasingleelementwhentheyaresu cientlyfarfromthebody[34,32].

Thesecondtypeoferrorassociatedwithvorticityadvectingordi usingthroughtheboundaryistypicallyhandledbyposingout owboundaryconditions.Forincompressible owtheseareusuallycalledconvectiveboundaryconditions,whereasincompressible owthetermnon-re ectingboundaryconditionisoftenused.Anothertechniqueistoselectivelyapplydampinginaregionnearthecomputationalboundary.Methodsthatemploythistechniquevaryfromadhocspeci cationoflayerwidth,dampingstrength,etc.,totechniquesthattheoreticallyspecifythedampingparametersaccordingtoamodel.Anexampleistheperfectlymatchedlayer[1]forlinearwaveequations(includinglinearizedcompressibleEulerequations[12])thatusesanalyticalsolutionstothegovern-ingequationstoderivedampingtermsthatpreventre ec-tionofwavesfromtheinterface.Anothertechniquecalledsuper-grid[6]isbasedonananalogywithturbulencemod-eling–thatthee ectoftheturbulencemodelistomodelscalestoo netoberesolvedinthecomputationalmesh,whereasthee ectoftheboundaryconditionistomodelscalestoolargetoberesolvedinthecomputationaldomain.Afulldiscussionofthesetechniquesisbeyondthescopeofthispaper;wereferthereadertosomerecentreferencesforfurtherdetails[33,15,25,5].Thesetechniquesaredesignedtoremovevorticityfromthedomainassmoothlyaspossibletherebypreventingundesirablere ec-tionsoraliasing.Mostdonotaccountforthevelocityinducedbyvorticitythathasalreadyexitedthedomain(anon-locale ect).

Wepresenthereanalternativetechniquethatsharessomefeatureswiththesepreviousmethods,especiallythoseof[31,34,6].Itisbasedonamulti-domainapproachthatalsosharessomeoperationswiththemultigridmethodforsolvingellipticequations.We rstdescribethemethodinwords.Thebasicideaistoconsiderthedomainasembeddedinalargerdomainbutwithacoarsermesh.Thecirculationontheinner(smaller, ner)meshistheninterpolatedorcoarsi edontotheouter(larger,coarser)mesh.ThePoissonequationissolved(withzeroboundaryconditions)ontheouterdomain.ThissolutionistheninterpolatedalongtheboundaryoftheinnermeshandthePoissonequationissolved,withthe‘‘corrected’’boundaryvaluespeci ed,ontheinnermesh.

Similartothevortexmergingmethoddiscussedabove,anyexistingcirculationintheouterportionofthelargerdomainisretainedfromtheprevioustimelevel.Inthisway,weapproximatelyaccountforcirculationthathasadvectedordi usedoutoftheinnerdomain.Clearly,thesolutiononthecoarsermeshcontainsalargertruncationerrorfortheevolutionofthisvorticity.However,inversionoftheLaplacianisasmoothingoperation.Highfrequencycomponentsofthesolutioninducedbycirculationintheoutermeshdecaymorerapidlythanlow-frequencycompo-nents.Beinginterestedinthe owinthevicinityofthebody(anditswake),wediscardthesolutionsintheouterregiononlyretainingthevelocityitinducesontheinnerdomain.

Weapplythistechniquerecursivelyanumberoftimes,enlarging(andcoarsening)thedomainineachgridlevel.Wechoosetokeepthetotalnumberofgridpointsineachdirection xedoneachmesh;wemagnifythedomainandcoarsenthegridbyafactorof2ateachgridlevel.Thepro-cedureisshownschematicallyinFig.5.Thevorticityisrepeatedlycoarsi edoneachprogressivegrid.ThePoissonequationisthensolvedonthelargestdomain,inturnpro-vidingaboundaryconditionforthenextsmallerdomain.Theprocessisthenrepeateduntilwereturntotheoriginaldomain.

Thevelocity elddecaysalgebraicallyinthefar- eldandwethusexpecterrorsassociatedwiththeboundaryconditiononthelargestdomaintodecreasegeometricallyasthesizeofthelargestdomainisincreased.Intheworstcaseofatwo-dimensional owwithnon-zerototalcircula-tion,thevelocitydecayswiththeinverseofthedistancetothevorticalregion.AnalyticalestimatesgiveninAppendixBshowthatweobtainafactorof4reductioninthebound-aryerrorwitheachprogressivelylargergrid.This,ofcourse,iswhatwouldbeobtainedbysimplyextendingtheoriginalgridtoadistanceequaltotheextentofthelargestgrid,butduetothecoarseningoperation,thecostincreaseslinearlywithincreasingextent,ratherthanqua-dratically(intwodimensions)orcubically(inthreedimensions).

Themethodcanthusbewrittenasfollows.Wede nethedomainofeachgridasDðkÞ,k=1,2,...,Ng,wherek=1referstotheoriginal(smallest)gridandk=Ngreferstothelargestone.Wethende nethemulti-domaininverseLaplacian

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T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462139

~s¼SKÀ1Sc~

;ð28Þ

where~cisanarbitraryinputvector(withlengthequaltothenumberofdiscretecirculationvaluesonthegrid),~sisthesolution(withlengthequaltothenumberofdiscretestreamfunctionvalues),andtheoperatorSKSimpliesthefollowingoperations:

~cð1Þ¼~c8

;

ð29Þ>~cðkÞwherex2DðkÞnDðkÀ1Þ;~cðkÞ¼<

:

PðkÀ1Þ!ðkÞð~cðkÀ1ÞÞwherex2DðkÀ1Þ>;

ð30Þk¼2;3;...;Ng;~sðNgþ1Þ¼0;

ð31Þ~sðkÞ¼SKÀ1Sc~

ðkÞþbcs½Pðkþ1Þ!ðkÞð~sðkþ1ÞÞ ;k¼Ng;NgÀ1;...;1;ð32Þ~s¼SKSc~

¼~sð1Þ:ð33Þ

HereP(kÀ1)!(k)(k)!(kÀ1)isa ne-to-coarseinterpolationoperator

andPisitscoarse-to- necounterpartrestrictedtooDðkÀ1Þbybcs.

InconstructingP,itwouldbedesirabletopreserve(tomachineroundo )certainmomentsofthecirculationdis-tributionsothatthevelocitydecayratefarfromthebodyiscorrect.Inthepresentimplementation,weattempttopreserveonlythetotalcirculation.Switchingfrommatrix/vectortopoint-operatornotation,wewrite,forthetwo-dimensionalcase,

PðkÀ1Þ!ðkÞðc~

ðkÀ1ÞÞðkÀ1Þ

2i;2j¼~ci;jþ1ðkÀ1Þ1ðkÀ1Þ

2~ciÀ1;jþ2~ciþ1;j

þ12~cðkÀ1Þi;jÀ1þ12~cðkÀ1Þi;jþ1þ14~c

ðkÀ1ÞiÀ1;jÀ1þ14~cðkÀ1Þ1ðkÀ1Þiþ1;jÀ1þ4~ciÀ1;jþ1þ1ðkÀ1Þ4~c

iþ1;jþ1

:ð34ÞThe9-pointstencilleadstoaconservationofthetotalcir-culationandissecond-orderaccuratebasedonaTaylor-seriesexpansion.Wenotethatthecoe cientsinEq.(34)sumto4sincethecirculationinthe(dual)cellisthevortic-itymultipliedbythearea,andcoarsifyingthegridbyafac-torof2resultsinafactorof4increaseincellarea.Thethree-dimensionalversionofEq.(34)consistsofaveragingEq.(34)overtwoadjacent(i,j)planesofdatanormaltothevorticitycomponent,foreachofthethreecomponents.Forthecoarse-to- neinterpolationattheboundaryofthenext- nermesh,weusethevaluefromthecoarsermeshforthosegridpointsthatcoincide,andamid-pointlinearinterpolation(againsecond-orderaccurate)forthosepointsinbetween.

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.

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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=

0.02.

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5.2.PropagationofanOseenvortex

Inordertoevaluateerrorsassociatedwithvorticityadvecting/di usingthroughthecomputationalboundary,weagainusetheanalyticalsolutionassociatedwithanOseenvortex.Thevortexisinitializedat(x,y)=(0,0)andadvectedbyanotherwiseuniform owwithspeedequaltothemaximumvelocityofthevortex.ThevorticityandazimuthalvelocityarestillgivenbyEqs.(39)and(40),respectively,r¼q

butðxÀUtÞ2

þy2 withtheradius,rrede nedwith.

Again,Candtheinitialtime,t0aresetsothatatt=t0,themaximumspeedassociatedwiththevortexaloneisUandoccursatr=R.AgainwesetRe=300.

Fig.8showstheerrorinthevelocityattheoriginforadomainthatnominallyextendsto±5RwithD¼0:05.Sincethevelocitydecayslike1/r,ithasalong-rangeR

e ect.Toachievelessthan1%errorwithoutcorrectedboundaryconditions,thedomainwouldneedtoextendto±100R.Theerrorisinitiallyzero(evenwiththeuncorrectedboundaryconditions)duetosymmetry.Astimepro-gresses,theerrorincreasesandreaches25%forNg=1.Thisoccursasthevortexpropagatesthroughtherightboundaryofthedomain.WithNg>1,thevortexispro-gressivelytransferredtothenextlargestmeshatintervalsoftime5·2nÀ1,n=1,...,Ng.WithNg=5,theerrorstaysbelow1%uptonon-dimensionaltime80,whenitleavesthecoarsest,largestmesh.Therearesmalloscillationsintheerrorevidentduringgrid-to-gridtransfertimes.Theassociatedtotalcirculationchangesbyatmost5%duringthesetransfers.WithNg=10,errorfromtheboundaryconditionisundetectableuptotime100andtheerroriscontrolledbythesecond-orderdiscretizationerrorandstaysbelowabout0.2%.Thesolutionattime100isshowninFig.9onthelargestmesh.Themagni edregionisshownasinaninsetandshowscontoursofthevorticityandnormalvelocity.Bytime100,thevortexwouldhavephysicallydi usedtoacoresizeofabout1.6R,whereasthegridspacingonthelargestdomainis12.8R!Theveloc-ity eldnearthecoreiscompletelywrong,butthecircula-

tionisnearlyconservedandthisinducesthecorrectpotential owfarfromthecore.Thephysical(smallest)domainisalsodepictedontheplotand,asisshowninFig.8,theerrorattheoriginisstilllessthanaboutoneper-centofthecorrectvalueatthattime.5.3.Potential owoveracylinder

Asa nalexample,weconsiderthepotential owinducedatt=0+byanimpulsivelystartedcylinderofdiameterD.Theimmersedboundaryuses571equallyspacedLagrangianpointsandthedomainisde nedsnuglyaroundthebody,extendingto±0.55DineachdirectionwithgridspacingD=0.0055D.Weinitiateauniform

ow

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withspeedUandletthebody‘‘materialize’’att=0.Thesolutionisobtainedbyperforming1time-stepoftheNavier–Stokessolutionusingthefastmethodwithmulti-domainboundaryconditions.A ow eldobtainedwithNg=4ispresentedwiththeexactpotential owsolutioninFig.10.Thestreamlinesarefoundtobeinagreementwithaslightdi erenceneartheimmersedboundaryduetotheregularizednatureofthediscretedeltafunction.InFig.11,wecomparetheexactpotential owsolutiontothenumericalsolutionalongthetopboundaryoftheinner-mostdomainfordi erentNg.WeobservetheestimatedOð4ÀNgÞconvergenceÀ3(seeAppendixB)downtoalevelofabout10afterwhichtheleading-ordererrorisdomi-natedbythetruncationerrorarisingfromthediscretedeltafunctionsattheimmersedboundaryandthediscretizationofthePoissonequation.6.Performanceofthefastmethod

Weconcludebymeasuringtheperformanceofthefastnullspace/multi-domainimmersedboundarymethodcom-paredtotheoriginalperformancebytheIBPM.First,wesimulate owsoverastationarycircularcylinderofdiam-eterDandcomparetopreviouslypublishedresults[18,36].ComputationsareperformedonthedomainDð1Þ¼½À1;3 ½À2;2 withD=0.02DwhereNgisvariedbetween1and5.Thecylinderiscenteredattheorigin.The owisimpulsivelystartedatt=0,andthebodyissta-tionary.ThustheCholeskydecompositionisusedtosolveEq.(36).

Aftertransiente ectsassociatedwiththeimpulsively-started owhavediedaway,weexaminewakestructuresandforcesonthecylindersfromfordi erentvaluesofNg.ThesearecomparedwithpreviousresultsforRe=40and200inTables1and2,respectively.Forthesteady owatRe=40wereportcharacteristicdimensionsoftherecir-culationbubbleinthewake,andfortheunsteady owatRe=200,wereportsheddingfrequencyand uctuatingliftanddragcoe cients.CharacteristicdimensionsofthewakeareillustratedinFig.12.ItisevidentthatasNgisincreased,thefastmethodgivesnearlyidenticalresultstothepreviouslypublisheddata.ItappearsthatNg=4issuf-

Table1

Comparisonofresultsfromthefast-methodwithpreviouslyreportedvaluesforsteady-state owaroundacylinderatRe=40

l/d

a/db/dhCDSpeed-upRe=40

Present(Ng=2)1.690.600.5553.4°1.9225.8Present(Ng=3)2.010.670.5854.0°1.6818.5Present(Ng=4)2.170.700.5953.8°1.5814.2Present(Ng=5)2.200.700.5953.5°1.5511.3LinnickandFasel[18]2.280.720.6053.6°1.54–TairaandColonius[36]

2.30

0.73

0.60

53.7°

1.54

1

Table2

Comparisonofresultsfromthefast-methodwithpreviouslyreportedvaluesforunsteady owaroundacylinderatRe=200

St

CD

CLSpeed-upRe=200

Present(Ng=2)0.2061.47±0.049±0.66121.1Present(Ng=3)0.2001.40±0.052±0.7084.7Present(Ng=4)0.1971.36±0.046±0.7065.4Present(Ng=5)0.1951.34±0.045±0.6853.0LinnickandFasel[18]0.1971.34±0.044±0.69–TairaandColonius[36]

0.196

1.35±0.048

±0.68

1

cienttorecoverthepreviousresults.Notethatfortheori-ginalIBPM,computationsareperformedoveradomainof[À30,30]·[À30,30]by300·300stretchedgridpointswiththe nestresolutionofDx=Dy=0.02.ThetimestepforallcasesarechosentobeDt=0.01tolimitthemaxi-mumCourantnumberto1.

Inthetables,speed-upisde nedasthetimerequiredtocomputethelast50timestepsinthesimulationsnormal-izedbythetimeelapsedfortheoriginalIBPM.Bymeasur-ingthelast50timesteps,wegiveaconservativeestimateforspeed-upsincetheoriginalmethodisiterativeandtyp-icallyrequiresmanymoreiterationsforearliertimes.ThuswithNg=4thefastmethodreducesthecomputationaltimebyafactorofabout15forthesteady owand65fortheunsteady ow.Wehavefoundsimilarspeed-upsinavarietyofproblemsonwhichwehavetestedthecode.Wenotethatwehavethusfaronlyimplementedthefastmethodintwodimensions(theoriginalalgorithmhas

been

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validatedinbothtwoandthreedimensions).Speed-upsforthree-dimensionalproblemsarelikelytobemoredramaticasdiscussedinSection3.3.

Next,wecomparethespeed-upfromforatranslatingcircularcylindersimulatedbymovingtheLagrangianboundarypoints.NowEq.(36)issolvediterativelywiththeconjugate-gradientmethod.Acylinderoriginallyattheoriginatt=0isimpulsivelytranslatedtotheleftwithunitvelocitywithRe=200.TheinnermostdomainisselectedasDð1Þ¼½À5;1 ½À1;1 withD=0.02DandweuseNg=4multi-domains.Insidethishighlycon nedDð1Þ,thetranslatingcylindergeneratestwocounterrotat-ingvorticesinthewakeasshowninFig.13fort=3.5.Thevorticitypro leisinaccordwithpreviousresultsreportedin[36].Comparedtoacomputationperformedwiththeoriginalapproach,thepresentcomputationisfoundtobe43.4timesfaster.Recallthataspeed-upof53.0isobservedforastationarycylinder(Table2),whichsuggeststhattheoverallalgorithmisstillsolvede cientlyevenwithamovingimmersedboundary.7.Summary

Wehavereportedonimprovementstotheimmersedboundaryprojectionmethodfor owovertwo-andthree-dimensionalbodieswithprescribedmotion.Inprevi-ouswork[36],weshowedthattheIBmethodcanbeformulatedinanalgebraicallyidenticalwaytotheincom-pressibleNavier–Stokesequationswithoutanimmersedboundary.ThisformulationenablestheclassicalfractionalstepmethodtobeappliedtotheIBequations,eliminatingtheneedforanyconstitutiverelationforthemotionofthebody(andhenceassociatedsti ness),andensuringthattheno-slipanddivergence-freeconstraintsaresatis edtoarbi-trarilyhighprecision.Inthispaperweshowedthatthesolutioncanbesubstantiallyacceleratedbyemployinganullspace(discretestreamfunction)methodtosatisfyingthedivergencefree-constraint,andbyrestrictingthecom-putationtoequally-spacedmeshes.

Inthisfastmethod,theviscousterms,divergence-free,andno-slipconstraintsarestilltreatedimplicitly,butthelinearsystemsassociatedwiththePoissonequationandimplicitviscoustermscanbesolvedirectlywithfastsinetransforms.Inthesolution,thedivergence-freeconstraintisautomaticallysatis edtomachineprecision.Forstation-

arybodies,theno-slipconstraintcanalsobeenforcedtomachineprecisionbydirectsolutionoftheequationforthebodyforcesbyusingaCholeskydecomposition.Formovingbodies,iterativesolutionofthelinearsystemforthebodyforcesisstillrequired,butthesizeofthesystemisproportionaltothenumberofLagrangiansurfacepoints;thematrixispositivede niteandtheconjugate-gradienttechniqueise cientforitssolution.

NeartheIB,therestrictiontouniformmeshisastan-dardrequirementofthediscretedeltafunction;however,farfromthebody,thiswouldingeneralbeoverlyrestric-tiveasitisusefultostretchthemeshsothatthedomaincanbemadelargetoapproachthesolutiononanunboundeddomain.Wepursuedanalternativestrategyofimprovingthefar- eldboundaryconditionstothepointwherethedomaincanenclosethebody(andtheportionofthewakeonewishestoresolve)snugly.Wederivedamulti-domaintechniquethatsolvesthePoissonequationonprogressivelylarger,butcoarser,meshes.Vorticityisallowedtoadvectanddi usefrom nertocoarsermesh.Theresulting owontheoriginaldomainthenaccountsforboth(i)theslowlydecayingpotential owinducedbythebodymotion,and(ii)theslowlydecayinginducedvelocityassociatedwithvorticitythathasadvectedtolargedistancefromthebody.Whilethereiscostpenaltyassoci-atedwiththemulti-domainsolution,theoverallschemeemployingthefastnullspacemethodandmulti-domainboundaryconditionsisstillmorethananorder-of-magni-tudefasterthanouroriginalmethodintwodimensions.Thespeed-upresultsfrombene tsassociatedwiththefastnullspacemethodaswellasbeingabletousemorecom-pactdomains.

Thefastnullspacemethodandmulti-domainboundaryconditionsareequallyvalidforbothtwo-andthree-dimensional ows.Two-dimensionaltestcasesincludingstationaryandadvectingOseenvorticesand owoverimpulsively-startedcylindersdemonstratetheaccuracyofthemulti-domainboundaryconditions.WenotethatthetechniquesaregenerallyapplicabletotheincompressibleNavier–Stokesequationsonunboundeddomainswithorwithoutimmersedboundaries.Themulti-domaintech-nique,inparticular,shouldproveusefulinsimulating owsthatinvolveweakinteractionsof nite-circulationvorticesthathaveplaguedmethodsemployingperiodicorothersimpli edboundaryconditionsinthepast(e.g.[10,14]).

Acknowledgements

TheauthorsthankProf.BlairPerotfortheenlighteningdiscussionsonthefractionalstepmethod.Theexperimen-taldatapresentedinSection2.2weregenerouslysharedbyDr.WilliamDickson.ThisworkwaspartiallysupportedbytheUnitedStatesAirForceO ceofScienti cResearch(AFOSR/MURIFA9550-05-1-0369)andtheNationalScienceFoundation

(DMS-0514414).

一些ME专业提升的论文。

2144T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

AppendixA

Examplesofthepoint-operatornotationimpliedbysomeofthematrix–vectormultipliesinthepaperaregivenhere,forthe(relevant)caseofauniform,two-dimensional,staggeredgridwithequalgridspacing,D,inbothdirections(forwhichM=I).Theseoperatorscanallbesimplyderivedasspecialcasesofthoseusedforunstructuredmeshes[4],andthree-dimensionalversionsarestraightfor-ward.Notreportedherearetheregularization/interpola-tionoperators(E,H)whicharegivenby[36].Inwhatfollowsthesubscriptsiandjrefertotheithandjthcellsinthexandydirections,respectively,andthesuperscript(k),ifpresent,referstothekthcomponentofavectorquantitysuchasvelocityorgradientofpressure.ðDqÞð1Þ

ð1Þ

ð2Þ

ð2Þ

i;j¼qiþ1;jÀqi;jþqi;jþ1Àqi;j;

ð41ÞðGpÞð1Þ

i;j¼pi;jÀpiÀ1;j;

ð42ÞðGpÞð2Þ

i;j¼pi;jÀpiÀ1;j;ð43ÞðCsÞð1Þ

i;j¼si;jþ1Àsi;j;

ð44ÞðCsÞð2Þ

i;j

¼Àðsiþ1;jÀsi;jÞ;

ð45Þ

ðCTqÞ¼qð2Þ

ð2Þ

ð1Þ

ð1Þ

i;ji;jÀqiÀ1;jÀðqi;jÀqi;jÀ1Þ;

ð46Þ

ReD2ðLqÞðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

i;j¼qiþ1;jþqiÀ1;jþqi;jþ1þqi;jÀ1À4qi;j;k¼1;2:ð47Þ

AppendixB

Inthisappendix,wederivethetheoreticalestimatesfor

thevelocityerrorexpectedfromtheuseofmulti-domainboundaryconditionsforthePoissonequation.Wecon-siderthevelocityerrorsforpotential owaroundacylin-derandstationaryOseenvortexdiscussedearlierintheSection5.

B.1.Errorestimateforpotential owoveracylinderLetusconsideracircularcylinderofdiameterDsituatedattheorigininsidearectangulardomain[ÀL/2,L/2]·[ÀrL/2,rL/2].SidelengthLheredenotesthesizeofthesmallestDð1ÞandwerepresentNthesizeofthelargestdomainDðNgÞbyaL,wherea¼2g=2.Aspectratioofthedomainsisdenotedbyr.Toassessthevelocityerror,wecomputethevelocityerrorattopcenterofDð1Þðx¼

0;y¼r

LÞ.Otherpointsinthedomainscalesimilarly.Theexactpotential owsolutionatthispointforanunboundeddomainis

"u1þ D

2#

exact¼UL;ð48Þ

whereUisthefreestreamvelocityvalue.Thecorrespond-ingverticalvelocityviszeroatthispoint.

Toestimatetheerrorinducedbythemulti-domain

approach,thee ectofDirichletboundaryconditionsonthelargestdomaincanbeassessedusingthemethodofimages.Assumingthecylinderisplacedattheorigin,weobtain:X1uimages¼UþU

X1ðÀ1Þ

iþj

ðD=2Þ2ðy22

jÀxiÞ;

ð49Þ

i¼À1j¼À1

ðx2iþ

y2jÞ

wherexiandyjcorrespondstothedistancefromthecenter

ofÀthe(i,j)thÁcylindertothepointoferrorassessmentx¼0;y¼r

L.Substitutingxi=aLiandyandsubtractingtheexact(free-space)solution,j¼arLjþrLweobtain,theerror ¼uimagesÀuexact

1¼ÀUD2UD2XX1iþjr2Àjþ12a

2Ài2r2L2þ4a2L2ðÀ1Þi¼À1j¼À1r2jþ1222þi

UD2UD2p21

¼ÀXr2L2þ4a2L24ðÀ1Þjfcsch2½bjða;rÞ

j¼À1

þsech2½bjða;rÞ g;

ð50Þ

wherebjða;rÞ¼follows.Forthe4a

ð1þ2ajÞ.Thesumcanbebrokenupasj=0term,aTaylorseriesexpansionforlargeaisused.Forj50,1+2ajcanbereplacedby2ajforlargeaintheexponentials.Wethenobtain:

¼ÀUD2p2 1

L8a2

3

þCðrÞ!

þOðaÀ4Þ;ð51Þwherethesum

CðrÞ¼X

1ðÀ1Þj

csch2

prj þ2 prj !ð52Þ

j¼1

2sech2isindependentofaandcanbeevaluatednumericallyforagivenaspectratio,r.Thusweobtaintheestimate

$ÀUD2 1

L3þCðrÞ!4ÀNg;ð53Þwhichisthedesiredresultthatshowsthattheerrorde-creasesgeometricallywithincreasinggridlevel.Notethat

wecanbemoreprecisebynotingthatthesumC(r)goesrapidlytozeroforr>1,andincreaseslike1/r2forsmallr.Thuswecanalsowrite $

UD2½minðL;rLÞ

4ÀNg:

ð54Þ

B.2.ErrorestimateforstationaryOseenvortex

FortheOseenvortex,wefollowasimilarsetupastheprevioussection,butplaceanOseenvortexattheoriginofthedomain.ÀWenowconsiderthenormalvelocityerroratthepointx¼L

;y¼0Á.Assumingthattheboundaryoftheinnermostdomainisoutsidethevortexcore,wehave

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462145

vexact¼

C

2pðL=2Þ

;

ð55Þ

whereasthesolutionwithDirichletboundaryconditionsis

uCimages¼X1X1aLiþL2paðÀ1Þiþji¼À1j¼À1aLiþLrjÞ2;ð56Þþðasothattheerrorcanbewritten(aftersimilarsimpli cationstoanalysisintheprevioussection)

¼uimages(

Àuexact

¼C p pX12pÀ2þp

arcsch2arþarðÀ1Þi½cschðcþiða;rÞÞi¼1

þcschðcÀÞÞ

'

iða;rð57ÞwherecÆiða;rÞ¼2arð1Æ2aiÞ.Again,wecanexpandthetermsforlargea.Inthiscase,however,caremustbetakeninevaluatingthesumforlargeasincethesumiszeroto rstorderina.Weobtainasimilarresulttotheprevioussection,namely

$

C

minðL;rLÞ

4ÀNg:

ð58Þ

Afurthercancellationbetweenthesecondandthirdtermsontheright-handsideofEq.(57)occurswhenr=1.Wecanthenshowthat $

CÀNL

16g

whenr¼1:

ð59Þ

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