Computer Methods in Applied Mechanics and Engineering(13)

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

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