A Cartesian grid projection method for the incompr

ACartesianGridProjectionMethodfortheIncompressibleEulerEquationsinComplexGeometriesAnnS.AlmgrenJohnB.BellCenterforComputationalSciencesandEngineeringLawrenceLivermoreNationalLaboratoryLivermore,CA94550PhillipColellayTylerMarthaleryMechanicalEngineeringDept.UniversityofCaliforniaatBerkeleyBerkeley,CA94720ThisworkoftheseauthorswasperformedundertheauspicesoftheU.S.DepartmentofEnergybytheLawrenceLivermoreNationalLaboratoryundercontractNo.W-7405-Eng-48.SupportundercontractNo.W-7405-Eng-48wasprovidedbytheAppliedMathematicalSciencesProgramandtheHighPerformanceComputingandCommunicationsProgramoftheDOEOceofScienticComputingandbytheDefenseNuclearAgencyunderIACRO93-817andIACRO94-831.yResearchsupportedatUCBerkeleybytheUSDepartmentofEnergyOceofScienticComputingundergrantsFDDE-FG03-94-ER25205andFDDE-FG03-92-ER25140,andbybyaNationalScienceFoun-dationPresidentialYoungInvestigatorawardundergrantACS-8958522.1ABSTRACTManyproblemsinuiddynamicsrequiretherepresentationofcomplicatedinternalorexternalboundariesoftheow.Herewepresentamethodforcalculatingtime-dependentincompressibleinviscidowwhichcombinesaprojectionmethodwitha\Cartesiangridapproachforrepresentinggeometry.Inthisapproach,thebodyisrepresentedasaninterfaceembeddedinaregularCartesianmesh.TheadvectionstepisbasedonaCartesiangridalgorithmforcompressibleow,inwhichthediscretizationofthebodyneartheowusesavolume-of-uidrepresentation.Aredistributionprocedureisusedtoeliminatetime-steprestrictionsduetosmallcellswheretheboundaryintersectsthemesh.TheprojectionstepusesanapproximateprojectionbasedonaCartesiangridmethodforpotentialow.Themethodincorporatesknowledgeofthebodythroughvolumeandareafractionsalongwithcertainotherintegralsoverthemixedcells.Convergenceresultsaregivenfortheprojectionitselfandforthetime-dependentalgorithmintwodimensions.Themethodisalsodemonstratedonowpastahalf-cylinderwithvortexshedding.21IntroductionInthispaper,wepresentanumericalmethodforsolvingtheunsteadyincompressibleEulerequationsindomainswithirregularboundaries.Theunderlyingdiscretizationmethodisaprojectionmethod[22,5].Discretizationsofthenonlinearconvectivetermsandlaggedpressuregradientarerstusedtoconstructanapproximateupdatetothevelocityeld;thedivergenceconstraintissubsequentlyimposedtodenethevelocityandpressureatthenewtime.TheirregularboundaryisrepresentedusingtheCartesianmeshapproach[58],i.e.byintersectingtheboundarywithauniformCartesiangrid,withirregularcellsappearingonlyadjacenttotheboundary.TheextensionofthebasicprojectionmethodologytotheCartesiangridsettingexploitstheseparationofhyperbolicandelliptictermsofthemethodin[5]toallowustousepreviousworkondiscretizationofhyperbolicandellipticPDE’sonCartesiangrids.Thetreatmentofthehyperbolictermsisbasedonalgorithmsdevelopedforgasdynamics,andcloselyfollowsthealgorithmofPemberetal.[56,57].TheCartesiangridprojectionusesthetechniquesdevelopedbyYoungetal[72]forfullpotentialtransonicowtodiscretizetheellipticequationthatisusedtoenforcetheincompressibilityconstraint.Theoveralldesigngoalsofthemethodaretobeabletousethehigh-resolutionnitedierencemethodsbasedonhigher-orderGodunovmethodsfortheadvectivetermsinthepresenceofirregularboundariesthateectivelyaddapotentialowcomponenttothesolution.CartesiangridmethodswererstusedbyPurvisandBurkhalter[58]forsolvingtheequationsoftransonicpotentialow;seealso[71,40,62,72].Clarkeetal.[23]extendedthemethodologytosteadycompressibleow;seealso[30,29,28,52].ZeeuwandPowell[74],CoirierandPowell[24],Coirier[25],Meltonetal.[51],andAftosmisetal.[1]havedevelopedadaptivemethodsforthesteadyEulerandNavier-Stokesequations.Fortime-dependenthyperbolicproblems,theprimarydicultyinusingtheCartesiangridapproachliesinthetreatmentofthecellscreatedbytheintersectionoftheirregular3boundarywiththeuniformmesh.TherearenorestrictionsonhowtheboundaryintersectstheCartesiangrid(unlikethe\stairstepapproachwhichdenesthebodyasalignedwithcelledges),andasaresultcellswitharbitrarilysmallvolumescanbecreated.Astandardnite-volumeapproachusingconservativedierencingrequiresdivisionbythevolumeofeachcell;thisisunstableunlessthetimestepisreducedproportionallytothevolume.Althoughintheprojectionmethodconvectivedierencingisusedforthehyperbolictermsinthemomentumequation,webaseourmethodologyforincompressibleowontheexperiencegainedforcompressibleowinthehandlingofsmallcells.(Inaddition,wewillwishtoupdateotherquantitiesconservatively.)Themajorissues,then,indesigningsuchamethodarehowtomaintainaccuracy,stability,andconservationintheirregularcellsattheuid-bodyinterfacewhileusingatimesteprestrictedbyCFLconsiderationsusingtheregularcellsizealone.Werefertothisasthe\smallcellproblem.Noh[54]didearlyworkinthisareainwhichheusedbothcellmergingtechniquesandredistribution.LeVeque[42,43]andBergerandLeVeque[12]havedevelopedexplicitmethodswhichusethelargetimestepapproachdevelopedbyLeVeque[41]toovercomethesmallcellproblem.BergerandLeVeque[13,14]havealsostudiedapproachesinwhichthesmallcellproblemisavoidedbytheuseofarotateddierenceschemeinthecellscutbytheuid-bodyinterface.B