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.