2005_IMMERSED_BOUNDARY_METHODS

10Nov200414:25ARAR235-FL37-10.texAR235-FL37-10.sgmLaTeX2e(2002/01/18)P1:IBD10.1146/annurev.fluid.37.061903.175743Annu.Rev.FluidMech.2005.37:239–61doi:10.1146/annurev.fluid.37.061903.175743Copyrightc
2005byAnnualReviews.AllrightsreservedIMMERSEDBOUNDARYMETHODSRajatMittalDepartmentofMechanicalandAerospaceEngineering,GeorgeWashingtonUniversity,Washington,D.C.20052;email:[e-mail]erforTurbulenceResearch,MechanicalEngineeringDepartment,StanfordUniversity,Stanford,California94305;email:[e-mail]“immersedboundarymethod”wasfirstusedinreferencetoamethodde-velopedbyPeskin(1972)tosimulatecardiacmechanicsandassociatedbloodflow.ThedistinguishingfeatureofthismethodwasthattheentiresimulationwascarriedoutonaCartesiangrid,whichdidnotconformtothegeometryoftheheart,andanovelprocedurewasformulatedforimposingtheeffectoftheimmersedboundary(IB)ontheflow.SincePeskinintroducedthismethod,numerousmodificationsandrefinementshavebeenproposedandanumberofvariantsofthisapproachnowex-ist.Inaddition,thereisanotherclassofmethods,usuallyreferredtoas“Cartesiangridmethods,”whichwereoriginallydevelopedforsimulatinginviscidflowswithcomplexembeddedsolidboundariesonCartesiangrids(Berger&Aftosmis1998,Clarkeetal.1986,Zeeuw&Powell1991).Thesemethodshavebeenextendedtosimulateunsteadyviscousflows(Udaykumaretal.1996,Yeetal.1999)andthushavecapabilitiessimilartothoseofIBmethods.Inthisreview,weusethetermim-mersedboundary(IB)methodtoencompassallsuchmethodsthatsimulateviscousflowswithimmersed(orembedded)boundariesongridsthatdonotconformtotheshapeoftheseboundaries.Furthermore,thisreviewfocusesmainlyonIBmeth-odsforflowswithimmersedsolidboundaries.Applicationoftheseandrelatedmethodstoproblemswithliquid-liquidandliquid-gasboundarieswascoveredinpreviousreviewsbyAndersonetal.(1998)andScardovelli&Zaleski(1999).ConsiderthesimulationofflowpastasolidbodyshowninFigure1a.Theconventionalapproachtothiswouldemploystructuredorunstructuredgridsthatconformtothebody.Generatingthesegridsproceedsintwosequentialsteps.First,asurfacegridcoveringtheboundaries
bisgenerated.Thisisthenusedasaboundaryconditiontogenerateagridinthevolumefoccupiedbythefluid.Ifafinite-differencemethodisemployedonastructuredgrid,thenthedifferentialformofthegoverningequationsistransformedtoacurvilinearcoordinatesystemalignedwiththegridlines(Ferziger&Peric1996).Becausethegridconformstothesurfaceofthebody,thetransformedequationscanthenbediscretizedinthe0066-4189/05/0115-0239$14.00239Annu.Rev.Fluid.Mech.2005.37:239-261.Downloadedfromarjournals.annualreviews.orgbyShanghaiJiaoTongUniversityon09/27/05.Forpersonaluseonly.10Nov200414:25ARAR235-FL37-10.texAR235-FL37-10.sgmLaTeX2e(2002/01/18)P1:IBD240MITTAL
IACCARINOFigure1(a)Schematicshowingagenericbodypastwhichflowistobesimu-lated.Thebodyoccupiesthevolumebwithboundary
b.Thebodyhasacharac-teristiclengthscaleL,andaboundarylayerofthicknessδdevelopsoverthebody.(b)SchematicofbodyimmersedinaCartesiangridonwhichthegoverningequationsarediscretized.computationaldomainwithrelativeease.Ifafinite-volumetechniqueisemployed,thentheintegralformofthegoverningequationsisdiscretizedandthegeometricalinformationregardingthegridisincorporateddirectlyintothediscretization.Ifanunstructuredgridisemployed,theneitherafinite-volumeorafinite-elementmethodologycanbeused.Bothapproachesincorporatethelocalcellgeometryintothediscretizationanddonotresorttogridtransformations.NowconsideremployinganonbodyconformalCartesiangridforthissimu-lation,asshowninFigure1b.InthisapproachtheIBwouldstillberepresentedthroughsomemeanssuchasasurfacegrid,buttheCartesianvolumegridwouldbegeneratedwithnoregardtothissurfacegrid.Thus,thesolidboundarywouldcutthroughthisCartesianvolumegrid.Becausethegriddoesnotconformtothesolidboundary,incorporatingtheboundaryconditionswouldrequiremodifyingtheequationsinthevicinityoftheboundary.Preciselywhatthesemodificationsareisthesubjectofadetaileddiscussioninsubsequentsections.However,as-sumingthatsuchaprocedureisavailable,thegoverningequationswouldthenbediscretizedusingafinite-difference,finite-volume,orafinite-elementtechniquewithoutresortingtocoordinatetransformationorcomplexdiscretizationoperators.Atthispointitisusefultocomparetheadvantagesanddisadvantagesofthesetwoapproaches.Clearly,imposingtheboundaryconditionsisnotstraightforwardinIBmethodsand,furthermore,theramificationsoftheboundarytreatmentontheaccuracyandconservationpropertiesofthenumericalschemearenotobvious.Inaddition,alignmentbetweenthegridlinesandthebodysurfaceinbody-conformalgridsallowsbettercontrolofthegridresolutioninthevicinityofthebodyandthishasimplicationsfortheincreaseingridsizewithincreasingReynoldsnumber.Forexample,considerthecaseofthetwo-dimensional(2D)bodyofcharacteristiclengthLinFigure1awithaboundarylayerofthicknessδwhereinitisrequiredAnnu.Rev.Fluid.Mech.2005.37:239-261.Downloadedfromarjournals.annualreviews.orgbyShanghaiJiaoTongUniversityon09/27/05.Forpersonaluseonly.10Nov200414:25ARAR235-FL37-10.texAR235-FL37-10.sgmLaTeX2e(2002/01/18)P1:IBDIMMERSEDBOUNDARYMETHODS241toprovide