Direct Simulation Monte Carlo for Atmospheric Entry.docx
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DirectSimulationMonteCarloforAtmosphericEntry
DirectSimulationMonteCarloforAtmosphericEntry
2.CodeDevelopmentandApplicationResults
IainD.Boyd
DepartmentofAerospaceEngineering
UniversityofMichigan
AnnArbor,Michigan,USA
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Abstract
ThedirectsimulationMonteCarlomethod(DSMC)hasevolvedover40yearsintoapowerfulnumericaltechniqueforthecomputationofcomplex,nonequilibriumgasflows.Inthiscontext,nonequilibriummeansthatthevelocitydistributionfunctionisnotinanequilibriumformduetoalownumberofintermolecularcollisionswithinafluidelement.Inatmosphericentry,nonequilibriumconditionsoccurathighaltitudeandinregionsofflowfieldswithsmalllengthscales.Inthissecondarticleoftwoparts,severaldifferentimplementationsoftheDSMCtechniqueinvarious,widelyusedcodesaredescribed.ValidationoftheDSMCtechniqueforhypersonicflowsusingdatameasuredinthelaboratoryisdiscussed.AreviewisthenprovidedoftheapplicationoftheDSMCtechniquetoatmosphericentryflows.IllustrationsofDSMCanalysesareprovidedforslenderandbluntbodyvehiclesforentryintoEarth,followedbyexamplesofDSMCmodelingofplanetaryentryflows.
1.0Introduction
ThedirectsimulationMonteCarlo(DSMC)methodwasfirstintroducedbyGraemeBirdin1961[1].Sincethattime,Birdhaswrittentwobooksonthemethod[2,3]andthousandsofresearchpapershavebeenpublishedthatreportondevelopmentandapplicationofthetechnique.TheDSMCmethodismostusefulforanalysisofkineticnonequilibriumgasflows.Inthiscontext,nonequilibriumindicatesthatthevelocitydistributionfunction(VDF)ofthegasmoleculesisnotinthewell-understood,Maxwellian,equilibriumform.ThephysicalmechanismthatpushestheVDFtowardsequilibriumisinter-molecularcollisions,andsoagasfallsintoanonequilibriumstateunderconditionswherethereisnotalargeenoughnumberofcollisionsoccurringtomaintainequilibrium.Thetwomainphysicalflowconditionsthatleadtononequilibriumarelowdensityandsmalllengthscales.Alowdensityleadstoareducedcollisionratewhileasmalllengthscalereducesthesizeofafluidelement.TheusualmetricfordeterminingwhetheraparticulargasflowisinastateofnonequilibriumistheKnudsennumberdefinedasfollows:
(1.1)
whereλisthemeanfreepathofthegasandListhecharacteristiclengthscale.Themeanfreepathistheaveragedistancetraveledbyeachparticlebetweencollisionsandisgivenforahardsphereby
(1.2)
wherenisthenumberdensity,andσisthehardspherecollisioncrosssection.Thus,atlowdensity,themeanfreepath(andthereforeKn)becomeslarge.Similarly,forsmalllengthscales,LbecomessmallandagainKnbecomeslarge.Asaguidingrule,itisgenerallyacceptedthatkineticnonequilibriumeffectsbecomeimportantwhenKn>0.01.
Atmosphericentryflowconditionsmayfallintothekineticnonequilibriumregimeatsufficientlylowdensity(thatoccursathighaltitudeinaplanet’satmosphere)andforverysmallentryshapes(e.g.meteoroidsthathaveadiameterontheorderofacentimeter[4]).Inaddition,situationsarisewherelocalizedregionsofaflowmaycontainlowdensity(e.g.thewakebehindacapsule)orsmalllengthscales(e.g.sharpleadingedgesonavehicle,orshockwavesandboundarylayersthatmayhaveverysteepspatialgradientsinflowfieldproperties).AnalysisofhighKnudsennumberflowscouldinprinciplebeperformedthroughsolutionoftheBoltzmannequation,thatisthefundamentalmathematicalmodelofdilutegasdynamics[3].However,developmentofrobustandgeneralnumericalsolutionschemesfortheBoltzmannequationhasprovedasignificantchallenge.TheDSMCtechniqueemulatesthesamephysicsastheBoltzmannequationwithoutprovidingadirectsolution.TheDSMCmethodfollowsarepresentativesetofparticlesastheycollideandmoveinphysicalspace.IthasbeendemonstratedthatDSMCconvergestosolutionoftheBoltzmannequationinthelimitofaverylargenumberofparticles[3].
Inpartoneofthisarticle[5],thefundamentalaspectsoftheDSMCtechniquearefirstdescribedwithanemphasisonphysicalmodelingissuesrelatedtoitsapplicationtohypersonic,atmosphericentryproblems.Inthissecondarticle,thecapabilitiesoftheDSMCmethodareillustratedwithregardstosimulationofhypersonic,laboratoryexperimentsandthentoseveraldifferentvehicleentryapplications:
(1)Earthentryofslendervehicles;
(2)Earthentryofbluntvehicles;and(3)entryintoplanetaryatmospheres.ThesestudieswillillustratethattheDSMCtechniquehasbeenverifiedusingseveraldifferentsetsofentryflightdata.ThelevelofconfidenceintheaccuracyoftheDSMCtechniquehasreachedthestagewhereitisnowroutinelyemployedforpre-missiondesignandpost-missiondataanalysisofatmosphericentryflows.
2.0FundamentalAspectsoftheDSMCTechnique
2.1GeneralFeatures
TheDSMCtechniqueemulatesthephysicsoftheBoltzmannequationbyfollowingthemotionsandcollisionsofalargenumberofmodelparticles.Eachparticlepossessesmolecularlevelinformationincludingapositionvector,avelocityvector,andphysicalinformationsuchasmassandsize.Particlemotionandcollisionsaredecoupledoveratimestep∆tthatissmallerthanthelocalmeanfreetime.Duringthemovementofparticles,boundaryconditionssuchasreflectionfromsolidsurfacesareapplied.ThephysicaldomaintobesimulatedinaDSMCcomputationiscoveredbyameshofcells.Thesecellsareusedtocollecttogetherparticlesthatmaycollide.ThereareanumberofDSMCschemesforsimulatingcollisionsandallofthemachieveafasternumericalperformancethanthemoleculardynamics(MD)method[6]byignoringtheinfluenceoftherelativepositionsofparticleswithinacellindeterminingparticlesthatcollide.Thissimplificationrequiresthatthesizeofeachcellbelessthanthelocalmeanfreepathoftheflow.Bird’sNoTimeCounter(NTC)scheme[3]isthemostwidelyusedcollisionschemeinwhichanumberofparticlepairsinacellareformed.Eachofthepairsofparticlesisformedatrandomregardlessofpositionwithinthecell,andthenaprobabilityofcollisionforeachpairisevaluatedusingtheproductofthecollisioncrosssectionandtherelativevelocityoftheparticlepair.Thisprocedurereproducestheexpectedequilibriumcollisionrateunderconditionsofequilibrium.AnumberofcollisioncrosssectionmodelshavebeendevelopforDSMC,withthemostwidelyusedformsbeingtheVariableHardSphere(VHS)[7]andtheVariableSoftSphere(VSS)[8].Forhypersonicflow,VHSisconsideredsufficientlyaccurate.ValuesoftheVHSandVSScollisionparametersformanycommonspeciesareprovidedinBird[3].Itisdeterminedwhethertheparticlepairactuallycollidesbycomparingthecollisionprobabilitytoarandomnumber.Whenacollisionoccurs,post-collisionvelocitiesarecalculatedusingconservationofmomentumandenergy.FortheVHSmodel,isotropicscatteringisassumedinwhichtheunitvectoroftherelativevelocityisassignedatrandomontheunitsphere.
Thecellsemployedforsimulatingcollisionsarealsooftenusedforthesamplingofmacroscopicflowpropertiessuchasdensity,velocity,andtemperature.Thereisnonecessitytohavethecollisionandsamplingcellsbeidentical,however,andsometimesacoarsermeshisusedforsampling.
ThebasicstepsineachiterationoftheDSMCmethodare:
(1)moveparticlesoverthetimestep∆t;
(2)applyboundaryconditionssuchasintroducingnewparticlesatinflowboundaries,removingparticlesatoutflowboundaries,andprocessingreflectionsatsolidboundaries;(3)sortparticlesintocellsandcalculatecollisions;and(4)sampleaverageparticleinformation.Asimulationwillbeginfromsomeinitialcondition,anditwillrequireafinitenumberofiterationsfortheflowtoreachasteadystate.Generally,steadystateisdetectedasalevelingoffofthetotalnumberofparticlesinthesimulation.Aftersteadystateisreached,thesimulationiscontinuedafurthernumberofiterationsinordertoreducethestatisticalnoiseinthesampledinformationtoanacceptablelevel.AtypicalDSMCcomputationmayemployonemillionparticles,reachsteadystateafter50,000iterations,andcontinuesamplingforafurther50,000iterations.Onamoderndesktopcomputer,suchasimulationshouldtakeabout3hours.
WhiletheideasbehindtheDSMCtechniquearesimple,implementationinanalgorithmtakesonmanydifferentforms.SpecificDSMCalgorithmshavebeendevelopedforvectorcomputers[9]andparallelcomputers[10,11].Birdhasfocusedworkoncustomizingthealgorithmtoachieveefficientperformanceonsingleprocessormachines[12].
HavingprovidedageneraloverviewofthebasicelementsoftheDSMCmethod,thereaderisreferredtothefirstarticle[5]whereamoredetailedreviewisprovidedofthephysicalmodelsusedinDSMCthataremostcriticaltotheanalysisofhypersonicentryflows.
2.2DSMCCodes
UnlikeCFD,therearearelativelysmallnumberofdifferentimplementationsoftheabovefundamentalDSMCideasinnumericalcodes.AmongthemostwidelyusedDSMCcodesforhypersonicentryanalysisareDS2V/3V[13],DAC[11],SMILE[14],andMONACO[10].Thesecodesvarymainlyinthetreatmentofcollisionselectionmethodsandmeshtopology(fromorthogonalcut-cellstobody-fittedunstructuredcells).Mostofthecodesareparallelizedandthree-dimensional.Resultsfromallofthesecodesforanalysisofhypersonicentryflowsareincludedinthefollowingsection.
DS2V/3VcontinuestoundergodevelopmentbyB