Remeshed smoothed particle hydrodynamics for the s

JournalofComputationalPhysics182,67–90(2002)doi:10.1006/jcph.2002.7152RemeshedSmoothedParticleHydrodynamicsfortheSimulationofViscousandHeatConductingFlowsA.K.Chaniotis,∗D.Poulikakos,∗andP.Koumoutsakos†∗InstituteofEnergyTechnology,LaboratoryofThermodynamicsinEmergingTechnologies,ETH,Z¨urich,Switzerland;†InstituteofComputationalSciences,ETH,Z¨urich,Switzerland;andCenterforTurbulenceResearch,NASAAmes,MoffettField,California94035E-mail:[e-mail]24,2001;revisedMay9,2002Wepresentanextensionoftheclassicalschemeofsmoothedparticlehydrody-namics(SPH)fortheaccuratehandlingofdiffusiontermsinthemomentumandenergyequationofviscousandheatconductingflows.AkeyaspectofthepresentSPHapproachistheperiodicreinitialization(remeshing)oftheparticlelocations,whicharebeingdistortedbytheflowmap.High-ordermomentconservingkernelsarebeingimplementedforthisremeshingprocedureleadingtoaccuratesimulations.TheaccuracyoftheproposedSPHmethodologyistestedforanumberofbenchmarkproblemsinvolvingflowandenergytransport.Theresultsdemonstratethatthepro-posedSPHmethodologyiscapableofDNSqualitysimulationswhilemaintainingitsrobustnessandadaptivity.c2002ElsevierScience(USA)1.INTRODUCTIONThesmoothparticlehydrodynamics(SPH)methodisaLagrangiannumericalmethodintroducedbyGingoldandMonaghan[1],inordertomodelproblemsincontinuumphysicswhilecircumventingsomeofthelimitationsofgrid-basedmethods.SPHisarobustnumer-icaltechniquethathasbeenappliedtoawiderangeofproblems,rangingfromcompressiblefluidmechanicstoastrophysicssimulationsandflowstructureinteractions[1–6].AlthoughthemethodenjoysthepropertiesofLagrangianschemes,suchasautomaticadaptivityandnumericalstability,theextensionofthemethodinordertohandlediffusion-typeeffectshasbeenlimited.OneofthekeydifficultiesisthehandlingofdiffusiontypeoperatorsontheLagrangianmesh,whichisusuallydistortedbytheflowmap.Amethodologytoovercomethesedifficultiesispresentedinthispaper.ThedevelopmentoftheclassicalSPHmethodologyforcompressibleflowfieldsisde-scribedindetailbyMonaghan[7].Thecomputationalelementsareparticleswhoselocation670021-9991/02$35.00c2002ElsevierScience(USA)Allrightsreserved.68CHANIOTIS,POULIKAKOS,ANDKOUMOUTSAKOSisfollowinginaLagrangianfashiontheflowmap.Theinitialflowfieldquantitiesarein-terpolatedontheparticlelocations,andalltheflowquantitiescanbereconstructedbyalinearsuperpositionoftheflowquantitiescarriedbytheparticlesasweighedbyasmoothinterpolationkernel[8,9].Thediscreteequationsareobtainedfromcontinuumequationsbyexpressingtheflowquantitiesasalinearsuperpositionofthephysicalquantitiesthatarebeingcarriedbytheparticles.SPHbelongstoaclassofLagrangianmethodscalledparticlemethods.Thekeyadvantageofallparticlemethodsistoavoidtheexplicitdiscretizationofthenonlinearconvectiontermwhilemaintaininganautomaticadaptivityforthecomputationalelements.However,particlemethodsarefacedwithdifficultieswhendealingwiththeapproximationofviscouseffects.Theapproximationofdiffusionoperatorsinthecontextofparticlemethodsisasubjectthathasbeenextensivelyaddressedinthecontextofvortexmethodsinthelastdecade[10].Severaloptionshavebeenidentifiedsuchasthederivationofconservativeschemesbasedontheapproximationofthediffusionoperatorbyanintegraloperator[11,12],thedifferentiationofthesmoothingkernel[13],andtheapproximationofthediffusionoperatoronthedistortedLagrangiangridusingsomeaveragingprocedures[14].Inthiswork,wepresentanumericalschemetoaccountfordiffusioneffectsinthecontextofsmoothparticlehydrodynamicsbyincorporatingaremeshingstrategyfortheparticlelocationsalongwithanefficientcalculationofthediffusionoperatorsinthedistortedparticlelocations.ThefactthatviscosityplaysanimportantroleinmanyphysicalphenomenaofengineeringinterestunderlinestheneedtoimprovethemodelingofviscousforceswhilemaintainingtheadaptivityandrobustnessofSPH.AcommonlyemployedmethodologytoaccountfordiffusioneffectsinSPHistointroduceanartificialviscositysothatconservationofmomentumisensured.However,thisschemeusuallyyieldsinaccurateresults,becausetheseparationofshearandbulkviscosityisnotallowed[7,15].AnalternativeapproachtoremedythissituationisbasedonaTaylorseriesexpansionofthefieldquantitiesintheneighborhoodofeachparticleandcombinesthestandardfirst-orderSPHderivativeswiththefinitedifferencesmethod[16].ThemethodofBrookshaw[16]iscomputationallyefficient,sinceonlythefirstderivativeofthekernelisrequiredandcon-servesthelinearmomentum,whiletheangularmomentumisapproximatelyconserved.Thistypeofapproximationforthediffusiontermhasbeenimplementedsuccessfullytosimulateheatconductionproblems[6,16]andincompressibleviscousflows[5]withsolidboundariesbutitmayleadtoinaccurateresultswhenthevelocityorthedensityfieldisnoisy[15].AnotherapproximationofviscouseffectsinSPHinvolvesthenestedapplicationofthedifferenceapproximationthuscalculatingsecond-orderderivativesfromfirstderivatives[15,17].Thismethodcancalculateanysecondderivativeinsuchawaythattheformulasaresymmetricandtheyconservethelinearandangularmomentum.However,thismethodisnotcomputationallyefficientasitwouldimplyrepeatedcalculationsinvolvingallparticlesandmi