Computer Methods in Applied Mechanics and Engineering(9)

0.02.

一些ME专业提升的论文。

T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–21462141

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

一些ME专业提升的论文。

2142T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

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

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