VOF方法

Thevolume-of-fluidmethod•1Basicproperties•2Interfacereconstruction•3Volume-of-fluidadvectionalgorithms1.BasicpropertiesfToidentifywhetheragivenfluidiispresentataparticularlocation,weuseaHeaviside(st-ep)functionH,definedbyx12ifxisinfluidi;ifxisnotinfluidi1,()0,()1();iHAsHHxxx,01(,)ijVDHHtfHxydxdyxHDtyuAstheinterfacemoves,theshapeoftheregionoccupiedbyeachfluidchanges,buteachfluidparticleretainsitsidentity.Thus,thematerialderivative(followingthemotionofafluidparticle)ofHiszero.0()00ffffttuuuinte0,,1,rfac01e,IffthenthecellisfilledwithonefluidiffthenthecellisfilledwithacutthecenotherflullFiscalliedidVfftolhenthumeofeFluidLow-orderVOFmethodsdonotneedtospecifythelocationoftheinterfaceinthetransitionregion,butageometricalinterpr-etationofthesemethodsshowsthatintwodimensionstheinter-facelineineachmixedcellisrepresentedbyasegmentparalleltooneofthetwocoordinateaxes.Reconstructionoftheinterfaceshape(2)Advectionofthereconstructedinterf-aceina(1givenvelocityfield.)twostepsofVOFmethod2.Volume-of-fluidinterfacereconstructionalgorithmsAllofthealgorithmsdescribedbelowexceptforSLICalsoreturnaslope,orequivalently,avectornnormaltotheinte-rface.Inthisarticleweadopttheconventionthatalwayspointsawayfromn,thedarkfluid.Thenormalvectorinthei;jthcelltogetherwiththevolumefractionfi;juniquelydeterminestheapproximatelinearinterfaceinthatcell.Thus,sincethevolumefractionfi;jisijngiven,allofthealgorithmsdescribedbelow(butSLIC)aresimplyrulesfordeterminingaunitnormalvectorfromthevaluesofthevolumefractionsinsomeneighborhoodofthei;jthcell.nxyxnydymxbThisVOFmethodsdonotneedtospecifythelocationoftheinterfaceinthetransitionregion,butageometricalinterpr-etationofthesemethodsshowsthatintwodimensionstheinter-facelineineachmixedcellisrepresentedbyasegmentparalleltooneofthetwocoordinateaxes.2.1.SimpleLineInterfaceCalculation2.2.Thecenterofmassalgorithmoneconsidersthedarkfluidtohaveamassdensityof1,andthelightfluidtohavenomass.Todeterminetheap-proximateinterfaceinthecenterofa(33)blockofcellsonefirstdeterminesthecenterofmass(x,y)ofthe(33)pointtothecenterofthecentercell.Thisvectoristakenastheunitnormaltotheapproximateinterface.nIfthecenterofmassalgorithmreproducesalllinesexactly,theninparticularforarbitrarymitmustreproducetheliney=mxexactly.theslopeisnotequaltom2.3.ThecentraldifferencealgorithmInthisalgorithm,onefindstheslopemoftheapproximateinter-facetakinghalfthedifferenceoftherightandlefthandcolumnsumsofthevolumefractions.11,1,11m2kijkijkkff1,1beanapproximationtotheycoordinateof1theinterfaceat(),Inthecentraldifferencealgorithm2wearedeterminingtheslopeoftheapproximateinterfacebytakingacenteredkiijkkiletyhfxihdifferenceofthediscretevariableyiInpracticeonedoesnotknowthatthetrueinterfaceisavalidfunctionofx(e.g.,itcouldbeaverticalline).Wecanaddressthisproblembyalsodetermininganapproximationmtombydifferencingtheytopandbottomrowsandchoosingthebestvalueofm.Onewaytochoosebetweenthetwovaluesistochoosethesmallervaluemmin{m,m}xy2.4.ParkerandYoungs’methodonecalculatesanapproximationtof,whichistakentopointinthedirectionnormaltotheapproximateinterface.22EWNSffffxfyf1,11,1,11,11,1,11,1,11,11,1,11,11()1()1()1(2)222ijijijijijijijijijijiENjWjiSffffffffffffffff2ParkerandYoungsreportthatseemstogivethebestresults.reconstructthislineexactlydoesn'treconstructthislineexactly2.5.EfficientLeastsquaresvolume-of-fluidinterfacereconstructionWeconsideragaintheheightfunctiony=f(x),orx=g(y)andfortheslopemwecomputethecenteredschemeandalsothebackwardandforwardscheme.11222,,11E(n)=h(),ljkiklklkiljffL2,,E(n)=hmax(),nklklfmfL3Volume-of-fluidadvectionalgorithmsOperatorsplitadvection,,11,,221**,,11,,22()()nnnijijijijnijijijijffFFffGtxtxG