Computer Methods in Applied Mechanics and Engineering(3)

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]

:

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.

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