Computer Methods in Applied Mechanics and Engineering(5)

一些ME专业提升的论文。

2134T.Colonius,K.Taira/Comput.MethodsAppl.Mech.Engrg.197(2008)2131–2146

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