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PRESSURE-VELOCITY EQUILIBRIUM HYDRODYNAMIC MODELS

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PRESSURE-VELOCITY EQUILIBRIUM HYDRODYNAMIC MODELSPRESSU

PRESSURE-VELOCITY EQUILIBRIUM

    HYDRODYNAMIC MODELS

    AvailableonlineatWWW.sciencedirect.corn

    .,

    ?ScienceDirect

    ActaMathematicaScientia2010,30B(2):563594

    数学物理

    http:?acts.wipm.ac.an

    PRESSUREVELOCITYEQUILIBRIUM

    HYDR0DYNAMICMoDELS

    DedicatedtoProfessorJamesGlimmontheoccasionofhis75thbirthday

    JohnGrove

    ComputationalPhysicsGroupComputer,Computational,andStatisticalSciencesDivision, CCS-2MSD413,LosAlamosNationalLaboratory,LosAMmos,NM87545

    E-mail:Jgrove@lan1.gov

    AbstractThisarticledescribesmathematicalmodelsforphaseseparatedmixturesof materialsthatareinpressureandvelocityequilibriumbutnotnecessarilytemperature equilibrium.Generalconditionsforconstitutivemodelsforsuchmixturesthatexhibita singlemixturesoundspeedarediscussedandspecificexamplesaredescribed. KeywordsEulerequations;multiplephasemixtures;non-equilibriumtemperaturemix-.

    tures

    2000MRSubjectClassification35Q31;76N15;76A02;76J20;76L05;76T30 1Introduction

    Amajorareaofcomputationalfluiddynamicsisthetreatmentofmaterialmixtures.In thisarticlewewil1discusssomeofthemathematicalconsequencesofmultiplematerialmixture

    models,inparticularmodelsthatare"close"tosinglematerialformulationsinthesensethat

    themodelisdescribedbyasinglemixturepressure,velocity,andsoundspeed.Suchmodels areusefulduetotheirwell-posedness,andtheirabilitytobefittedintoexistinghydrocode implementations.Themostpopularoftheseisthepressure

    temperature-velocityequilibrium

    model,whichassumesthatthematerialcomponentsinacomputationalcellarephaseseparated

    andinpressureandtemperatureequilibriumwithacommonvelocity.Traditionallythismode1

    isreferredtoasapressure-temperature(PT)equilibriummodel,withvelocityequilibrium

    understood.Howeverfortimescalesdominatedbyshockwaveinteractions,theP

    Tequilibrium

    assumptiontendstobeoverlydiffusive,andmixturemodelsthatrelaxthisassumptionare needed.Inthisarticlewilldiscussthemathematicalstructureofmodelsobtainedbyrelaxing therequirementoftemperatureequilibrium,whilemaintainingphaseseparationandpressure

    andtemperatureequilibrium.

    ReceivedJunuary6,2010.ThisworkswassupportedbytheLosAlamosNationalLaboratory ,

    anaffir.

    mativeaction/equalopportunityemployer,operatedbyLosAlamosNationalSecurity,LLC,fortheNational

    NuclearSecurityAdministrationoftheU.S.DepartmentofEnergyundercontractDE-AC52.06NA25396.

    564ACTAMATHEMATICASCIENTIAVbI.30Ser.B

    Computationaltreatmentsformultiplematerialflowscanberoughlybrokenupintothree categories,interfacetreatmentsthatattempttoresolvethematerialseparationsbyexplicitly trackingtheboundariesbetweenseparatecomponents(suchasexplicitgeometricfronttracking

    [5,9,10,11,12,13,14,17orlevelsetmethods[2,22,23,341),mixedcelltreatmentsforthe

interactionsbetweenmaterials[4,andhybridtreatmentssuchasthevolumeoffluidmethod

    20,26,38,39

    thatusebothmixedcellmodelstogetherwithreconstructionsoftheinterfaces betweenmaterials.Ideallyal1threemethodscanbecombinedintoonecomputationalsystemto

    provideboththehighfidelityinterfacerepresentationprovidedbytrackingwiththerobustness

    ofthemixedcelltreatments.

    Thechoiceofamixedcellmode1isextremelyproblemdependent.andmustbebasedon thelengthandtimescalesappropriateforagivenapplication.Forexampletheassumption ofpressure-temperaturevelocityequilibriumforamixturemightbebasedonthefollowing assumptions:

    1)Themicrostructureofthemixtureconsistsofvolumetricallydistinctcomponents. 2)Materialcomponentsareseparatedbyinterfaces/contactdiscontinuitiesacrosswhich pressureandtheinterfacialnormalcomponentofvelocityarecontinuous. 3)Surfacetensionbetweencomponentsinthemicrostructureisnegligible(nocapillarity duetothemicrostructure1.

    4)Shearacrossthemicrostructureinterfaceisnegligible(commonvelocity).Theassump

    tionbasicallyassertsthatthemicrostructureconsistsofmaterialcomponentsthatare"well mixed",eitherasseparateblobsofmaterial,orconvolutedinterfacesbetweenthecomponents.

    5)Theapplicationtimescalesaresufficientlylongthatthecomponentshavetimetocome intothermalequilibriumduetoan

    modeledprocessessuchasthermalconduction(common

    temperature).

    Ourmaininterestinthisarticleistoinvestigatethemathematicalstructureofmodelsthat relaxthislastcondition,sothatthemicroscopicallyseparatedcomponentsarenotrequiredto beintemperatureequilibrium.Morecomplicatedmodelsthatallowmultiplepressuresand/or

    velocitiesinthemicrostructurearealsoofgreatinterestbutarebeyondthescopeofthe

    modelsconsideredhere.Oneoftheaimsforthissetofmodelsaretoproduceequationsthat are"close".tothePTequilibriummodelandthusaresuitabletoberetro

    fittingintoexisting

    PTequilibriumcodeimplementations.

    ThegroupofSaurelet.a1.haspublishedanextensivesetofarticlesdescribingtwocompo- nentmixtures.TheseworksincludeGodunovschemesforpressure

    relaxationmodelssimilarto

    theBaer

    Nunziato[3]multiphasedetonationmodel[27,28,discretizedformsofthismodel[1j

    extensionstoturbulentflows[28,30],applicationstoheterogeneousexplosives[61,incorporation

    ofcapillarityeffectsinthemodel[24],evaporativefronttreatments

    16],shockjumprelations

    [8,31],relaxation-projectionschemes[25,29,metastablefluidmodels[32],andefficients

    olu-

    tionschemesforthesetypemodels[33].Inallofthesemodels,thegenerallimitoftheflow inthecaseofinfiniterelaxationisasinglepressuremodelwithpossiblymultiplecomponent temperatures.Inmanycasestherelaxationparametersaretreatedasnumerical''knobs",and often(butnotalways)thesolutionofinterestisthelimitunderinfiniterelaxationrates. Aswewillsee,suchsinglepressuremodelshaveaninfinitesetofpossibleclosurerelations, eachcorrespondingtopossiblydifferentflowphysicsofthemixture.Wewillshowthatfour No.2J.W.Grove:PRESSURE

    VELOCITYEQUILIBRIUMHYDRODYNAMICMODELS565

    basicmodelsincommonusagecanbeincludedinthisclass,pressure-temperatureequilibrium,

    volume-temperatureequilibrium,entropyadvection(thermalisolation),andvolumefraction

    advection(uniformstrainoruniformcompression).Additionalmodelscanbebuiltoutofthis setbyassumingtheflowhasthenatureofa''mixtureofmixtures",forexamplecomponentsare

themselvesmixturesofmaterialsinpressure

    temperatureorvolume-temperatureequilibrium.

    Wewillalsoexaminewhatconsequencestheflowmodelassumptionhasontheshockwave structureofthemateria1.Thisisnon-trivialsincegenerallythefullmodelisnotinconserva- tiveformandconsequentlytheRankine-Hugoniotequationforshocksisunder

    determined.As

    anexamplewewilldiscussapossibleapplicationofsuchmixturemodelstoradiationhydrody

    namics.Anothermajorgoalofthisworkwillbetotreatthemultiplematerialmixturemodels forgeneralequationsofstate,includingthepossibiltyofphasesthatundergophysicalphase changes.

    2ThermodynamicPreliminaries

    ThenotationweuseisthatofMenikoff

    [37byA.S.Wightmanalsoisveryhelpful

    andPlohr[18].TheintroductiontobookofIsrael

    inthethermodynamicdiscussion.Thesearticles

    containanumberofusefulthermodynamicidentitiesthatwewillusefreely.Weassumethat eachmaterialinthemixtureisgovernedbyaseparatethermodynamicallyconsistentequation ofstate,specificallyweassumetheexistenceofaC,piecewiseC,convexspecificinternal energye=e(S)foreachspecies(insubsequentsectionswewilldistinguishtheseparate specificinternalenergyfunctionsbyasubscriptforeachmaterial,herewesuppressthesubscript

    forclarityofnotion)asafunctionofspecificvolumeV=1(Pisthemassdensity)andspecific entropy^5,intheinteriorofaconvexdomain(S)?f~v,s{(s)fv>0,s?0),andthat

    e(S)islowersemicontinuousattheboundaryaQs.Sincethespecificinternalenergyis convex,lowersemicontinuityofeisequivalenttothestatementthatatanypointoilaQs, eiseithercontinuousorblowsupasitapproachestheboundaryfseeNiculescuandPersson [21).Thetemperatureandpressureofthematerialisgivenbythefirstlawofthermodynamics relation:

de=TdSPd(1)

    Thus=fandP=fs.Inaddition,weassumethatforfixedspecificvolume,the specificinternalenergyisamonotoneincreasingfunctionofspecificentropy,thusT?0and

    wecaninverttherelatione=e(S)toobtainaCconcaveentropyfunctionS(e)with convexdomainQe.NotethattheHessiansofe=e(S)andS=s(e)arerelatedbythe formula:

    (2)

    SincestrictconvexityisequivalenttothestatementthattheHessianmatrixispositivedefinite

    ,

    itisclearthatSisstrictlyconcaveat(e(s))ifandonlyifeisstrictlyconvexat((e)). TheequivalenceoftheconcavityofS(V,e)andtheconvexityofe(V,S)followsimmediately

    01

    

    P

    

    S,

    P1

    旦跳叫锻一:

    566ACTAMATHEMATICASCIENTIAVb1

    .3OSer.B

    frommonotonicityoftheentropy/energyrelationforfixedspecificvolume,sincefor0<&lt;

    e((1)+n,(1)5b+aS1)

    ifandonlyif

    (1O1)e(v0,So)+e(,S1),

    (1)S(Vo,eo)+aS(?,e1)S((1)+av0,(1)e0+ae1),

    wheree0=e(Y0,So)andel=e(?,S1).

    Thesoundspeedc,GrfineisenexponentF

    ,

    andthespecificheatatconstantvolume aredefinedby

    p2c2=

    

    r

    Ief

    OVSOVS'

    OTf02ef

    OVsOVOSI'

    TOTIaeI=_=_

    CvOSVOS2V

    Inregionswheree(S)istwicedifferentiable,convexityisequivalenttotheconditions:

    p2c.>0.

    .(

    >0'

    r)>0(4)

    Therelationbetweenpressureandtemperature,andspecificvolumeandspecificentropy

    canbeinvertedviatheLegendretransformation: G(P,)(si)n

    ?

    f

    n.

    {e(s)+PVTS}?

    Thequantityc(aT)istheGibb'Sfreeenergywithconvexdomain

    QP,={(T)IG(P,T)>o.)

    (5)

    SimilarlyonecaninverttherelationbetweenentropyandtemperaturetoderivetheHelmholtz

    freeenergy:

    withdomain

F(T)=,i{e(S)TS},SSl()6nv,

    s,

    QT={()lF(T)>oo}

    (6)

    ForaconvexCfunctione(),itcanbeshown(see[2lagain)thatG()isstrictly

    concave,uppersemi

    continuous(convex/concavefunctionsarealwayscontinuousintheinterior oftheirdomains),piecewiseC,andthatatlocationswhereG(T)isdifferentiable,itsatisfies therelation:

    dG=VdPsdT(7)

    No.2J.W.Grove:PRESSURE

    VELOCITYEQUILIBRIUMHYDRODYNAMICMODELS567

    FurthermoretheslopeofacurvealongwhichthepartialsofG(P'T)jumpsatisfythe ClausiusClapeyronequation:

    dPl?S

    dTIcoexAV(8)

    PressuresandtemperatureswheretheGibb'sfreeenergyderivativesjumpcorrespondtoco- existenceregionswherethematerialundergoesaphasetransition.Inequation(8),ASand AVdenotethechangeinthepurephasespecificentropyandspecificvolumeacrossthephase transition.

    Pointsonthecoexistencecurvecorrespondtoregionsofnonstrictconvexityforthespecific

    internalenergy.InparticulartheGibb'sfreeenergyiscontinuousacrossacoexistencecurve, sothatatapoint(P'T)onsuchacurvewehave:

    

    P?+?S=Ae

    AsbeforeAe,?and?Sdenotethechangeinthecorrespondingquantityacrossthecoex

    istencecurve.Equation(8)followsbydifferentiatingtheexpressionGdP,T)=Gg(T)with respecttotemperaturealongthecoexistencecurve,wherethesubscriptsdenotetheGibb'sfree

    energiesoneithersideofthecurve(oftenregardedasaliquidandgaseosphase).

    Thekeypointintheabovediscussionistheequivalenceoftheequationofstateformulain termsofaCpiecewiseCconvexspecificinternalenergyasafunctionofspecificvolumeand specificentropyandtheformulationofapiecewiseCstrictlyconcaveGibb'sfreeenergyas afunctionofpressureandtemperature(notetheGibb'sfreeenergymaynotbecontinuously differentiablealthoughleftandrightpartialderivativesalwaysexist).Indeedgivensucha Gibb'sfreeenergy,thespecificinternalenergyisrecoveredviatheLegendretransform: e(S)=sup{G(T)JP+TS}

    (P,)?QP,T

    (10)

    Itispreciselytheassumptionofconvexityandlowersemi-continuityofe(S)orstrictconcavity

    anduppersemicontinuityofG(T)thatimpliestheinvertabilityoftheLegendretransform betweenthespecificinternalandGibb'sfreeenergies(againseereference[21]fordetails). Wefinishthissectionbynotingthatthespecificentropyrelationasafunctionofspecific volumeandspecificinternalenergycanalsobeinverted.Indeed,since

    wecanformtheLegendretransform

    suP

    (e)Cnv.

    ds=de+PdV

    ,

    TS(e)eP=

    刍器(e()+PV-TS)

    G()一—一

    '

    Wecanthenrecovers(e)usingtheinverseLegendretransform

    s(e)=(in

    ?

    f

    QPIT

    e+PV-G(P,T)

()

    (12)

    (13)

    ACTAMATHEMATICASCIENTIAVb1.3OSer.B

    Theserelationsareimportantinestablishingtheuniquenessofthepressure-temperatureequi

    libriumsolutionbelow.

    Forlateruse,wealsodefinetheisothermalcompressibilityKT,theisentropiccompress

    ibilityKs,thecoefficientofthermalexpansion,andthespecificheatatconstantpressure Cpbytheformulas:

    =I==I,:IP'(14)

    InappendixAofMenikoffandPlohr18

    avarietyofusefulrelationsbetweenthesequantities

    arelisted.Inparticulartherelations:

    Ks

    =

    l=

    Cv

    ,

    I:r=,s=

    1

    CpKTCpITKTCvKT=l一一=.0:_=.fq一——KTW'(15)

    willproveusefulintheseque1.Thethermodynamicstabilityconditioncanbeexpressedin termsofthecompressibilitiesandspecificheatsas:

    Cp?Cv?0orj?Ks0

    3EquilibriumMixtures

    (16)

    Equilibriummixturescanbecharacterizedbytheexistenceofathermodynamicfreeen- ergyasafunctionoftwothermodynamicvariablesandparameterizedbythecomponentmass fractions.SuchfreeenergiesmightincludeaGibb'sfreeenergyasafunctionofpressureand

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