Perfectly matched layer as an absorbing boundary condition for the linearized Euler equations in

JOURNALOFCOMPUTATIONALPHYSICS

ARTICLENO.144,213–234(1998)CP985997

PerfectlyMatchedLayerasanAbsorbing

BoundaryConditionfortheLinearizedEuler

EquationsinOpenandDuctedDomains

ChristopherK.W.Tam,?LaurentAuriault,?andFrancescoCambuli?

?DepartmentofMathematics,FloridaStateUniversity,Tallahassee,Florida32306-4510;and?Dipartimento

diIngegneriaMeccanica,UniversitadegliStudidiCagliari,Piazzad’Armi,09123,Cagliari,Italy

E-mail:tam@math.fsu.edu

ReceivedOctober30,1997;revisedApril6,1998

Recently,perfectlymatchedlayer(PML)asanabsorbingboundarycondition

hasfoundwidespreadapplications.Theideawas?rstintroducedbyBerengerfor

electromagneticwavescomputations.Inthispaper,itisshownthatthePMLequations

forthelinearizedEulerequationssupportunstablesolutionswhenthemean?owhasa

componentnormaltothelayer.Tosuppresssuchunstablesolutionssoastorenderthe

PMLconceptusefulforthisclassofproblems,itisproposedthatarti?cialselective

dampingtermsbeaddedtothediscretizedPMLequations.Itisdemonstratedthat

withaproperchoiceofarti?cialmeshReynoldsnumber,thePMLequationscan

bemadestable.Numericalexamplesareprovidedtoillustratethatthestabilized

PMLperformswellasanabsorbingboundarycondition.Inaductedenvironment,

thewavemodesaredispersive.Itwillbeshownthatinthepresenceofamean

?owthegroupvelocityandphasevelocityofthesemodescanhaveoppositesigns.

ThisresultsinabandoftransmittedwavesinthePMLtobespatiallyamplifying

insteadofevanescent.Thusinacon?nedenvironment,PMLmaynotbesuitableas

c1998AcademicPressanabsorbingboundaryconditionunlessthereisnomean?ow.??

1.INTRODUCTION

Recently,Berenger[1,2]succeededinformulatinganabsorbingboundaryconditionforcomputationalelectromagneticsthathastheunusualcharacteristicthatwhenanoutgoingdisturbanceimpingesontheinterfacebetweenthecomputationdomainandtheabsorbinglayersurroundingit,nowaveisre?ectedbackintothecomputationdomain.Inotherwords,alltheoutgoingdisturbancesaretransmittedintotheabsorbinglayerwheretheyaredampedout.Suchalayerhascometobeknownasaperfectlymatchedlayer(PML).

Sinceitsinitialdevelopment,PMLhasfoundwidespreadapplicationsinelasticwavepropagation[3],computationalaeroacoustics,andmanyotherareas.Hu[4]wasthe?rstto

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Allrightsofreproductioninanyformreserved.

214TAM,AURIAULT,ANDCAMBULI

applyPMLtoaeroacousticsproblemsgovernedbythelinearizedEulerequations,linearizedoverauniformmean?ow.Hehassinceextendedhisworktononuniformmean?owandforthefullynonlinearEulerequations[5].FurtherapplicationsofPMLtoacousticsproblemsincludingwavemodesinductscanbefoundinthemostrecentworksofHuandco-workers[6,7].Inthesereferences,examplesareprovidedthatindicatethathighqualitynumericalsolutionscouldbefoundwithPMLusedasradiationorout?owboundaryconditions.

Inopenunboundeddomains,acousticwavesarenondispersiveandpropagatewiththespeedofsoundrelativetothelocalmean?ow.Insideaduct,thesituationiscompletelydifferent.Acousticwavesarerepeatedlyre?ectedbackbythecon?ningwalls.Forductswithparallelwalls,thecontinuousre?ectionoftheacousticwavesbythewallleadstotheformationofcoherentwavepatternscalledductmodes[8,9].Unliketheopendomain,ductmodesaredispersivewithphaseandgroupvelocitiesvarywithaxialwavenumber.Becauseofthedispersivenatureoftheductmodesmanyradiationboundaryconditionsthatworkwellinopendomainsareknowntobeinappropriateforductedenvironments.Forthisreason,Tam[10]inarecentreviewonnumericalboundaryconditionsforcomputationalaeroacousticssuggestedthattheboundaryconditionforaductedenvironmentberegardedasacategoryofitsown.

Therearethreeprimaryobjectivesinthiswork.First,weintendtoshowthatinthepresenceofamean?ownormaltoaPML,thestandardPMLequationsofthelinearizedEulerequationssupportunstablesolutions.EarlierTam[10]pointedoutthatthePMLequationswithmean?owhaveunstablesolutions.However,hedidnotshowthattheexistenceofinstabilitiesisduetothemean?owcomponentnormaltothelayer.Theoriginandcharacteristicsoftheseinstabilitiesareinvestigatedandanalyzed.Itisinterestingtomentionthatinhisearliestwork,Hu[4]reportedthathiscomputationencounterednumericalinstability.Butbyapplyingnumerical?ltering,hewasabletoobtainstablesolutions.Inlightofour?nding,webelievethatwhatHuencounteredwasnotinstabilityofhisnumericalschemebutthathisnumericalsolutioninadvertentlyexcitedtheintrinsicunstablesolutionofthePMLequations.NotdirectlyrelatedtotheinstabilityofthePMLequations,AbarbanelandGottlieb[11]recentlyanalyzedtheelectromagneticPMLequations.Theyconcludedthattheequationsareonlyweaklywell-posed.

Second,wewillshowthattheinstabilityisnotverystrong,namely,thegrowthratesaresmall.Alsotheinstabilitiesarecon?nedprimarilytoshortwaves.Itis,therefore,possibletosuppresstheinstabilitiesbytheadditionofarti?cialselectivedampingterms[12]tothediscretizedPMLequations.Itisimportanttopointoutthatarti?cialselectivedampingeliminatesmainlytheshortwavesandhasnegligibleeffectonthelongorthephysicalwaves.ThustheadditionofthesedampingtermsdoesnoteffecttheperfectlymatchedconditionsofthePML.

Third,wewillshowthataperfectlymatchedlayermaynotbesuitableasanabsorbingboundaryconditionforwavesinaducted?owenvironment.Themajordifferencebetweenacousticwavesinanopendomainandacousticwavesinsideaductisthatinanunboundedregionacousticwavesarenondispersivewhereasductmodesaredispersive.Itwillbeshownthatinthepresenceofamean?owthegroupandphasevelocityoftheductmodescanhaveoppositesigns.Becauseofthis,abandoftransmittedwaveswillactuallygrowspatiallyinsteadofbeingdampedinthePML.Inotherwords,thePMLequationsdonotdampthesewavemodesasabsorbingboundaryconditionoughttodo.Theexceptioniswhenthereisnomean?owintheduct.Inthisspecialcase,allthetransmittedwavesarespatiallydamped.

PMLFORLINEARIZEDEULEREQUATIONS215

InSection2,theuseofPMLforopendomainproblemsisdiscussed.ThestabilityofthePMLgoverningequationsisinvestigated.Itwillbeshownthattheadditionofdamp-ingtermstoformthePMLequationscanactuallycausethevorticityandacousticwavemodestobecomeunstable.ThesplittingofthevariablesinformulatingthePMLequationsleadstoahigherordersystemofequations.Thishighersystemsupportsextrasolutions.Theseextraorspurioussolutionsarefoundtobecomeunstablewhenthedampingcoef?cientislarge.NumericalexamplesareprovidedtoillustratethespreadoftheunstablesolutionfromthePMLbackintotheinteriorofthecomputationdomain.

InSection3,theeffectoftheadditionofarti?cialselectivedampingtermstothedis-cretizedPMLequationsisinvestigated.ItisshownthatwithanappropriatechoiceofmeshReynoldsnumber,theunstablesolutionsofthePMLequationscanbesuppressed.Numericalexamplesaregiventodemonstratetheeffectivenessofthemodi?edPMLasaradiation/out?owboundarycondition.

Section4dealswiththetheoryandapplicationofPMLtoductedinternal?owproblems.AneigenvalueanalysisiscarriedouttoshowtheexistenceofabandoffrequencyforwhichthePMLexertsnodampingontheacousticductmodes.ThesewavemodesactuallywouldgrowinamplitudeastheypropagatethroughthePML.Numericalresultsareprovidedtoillustratetheexistenceofthiskindofamplifyingductedacousticmodes.

2.OPENDOMAINPROBLEMS

LetusconsidertheuseofPMLasabsorbingboundaryconditionforthesolutionofthelinearizedEulerequations(linearizedoverauniformmean?ow)inatwo-dimensionalopendomainasshowninFig.1.Wewilluse??x=??y(themeshsize)asthelengthscale,a0x2(thesoundspeed)asthevelocityscale,

??asthetimescale,andρ0a0(whereρ0isthea0

FIG.1.TwodimensionalcomputationdomainwithPerfectlyMatchedLayersasboundaries.

216TAM,AURIAULT,ANDCAMBULI

meandensity)asthepressurescale.ThedimensionlessgoverningequationsinthePMLareformedbysplittingthelinearizedEulerequationsaccordingtothespatialderivatives.Anabsorptiontermisaddedtoeachoftheequationswithspatialderivativeinthedirectionnormaltothelayer.Forexample,forthePMLontherightboundaryofFig.1(notatthecorners)thegoverningequationsare[4]

???u1+σu1+Mx(u1+u2)+(p1+p2)=0?u2?+My(u1+u2)=0?v1?+σv1+Mx(v1+v2)=0?v2??+My(v1+v2)+(p1+p2)=0???p1+σp1+Mx(p1+p2)+(u1+u2)=0?p2??+My(p1+p2)+(v1+v2)=0,whereMxandMyarethemean?owMachnumbersinthexandydirections.σistheabsorptioncoef?cient.

Supposewelookforsolutionswith(x,y,t)dependenceintheformexp[i(αx+βy?ωt)].Itiseasyto?ndfrom(1)thatthedispersionrelationsofthePMLequationsare

??βMyαMx?1???2α2β2??=0βMyαMx?=0.(2)(3)(1)1?

Inthelimitσ→0,(2)and(3)becomethewell-knowndispersionrelationsoftheacousticandthevorticitywavesofthelinearizedEulerequations.

2.1.MeanFlowParalleltoPML

Dispersionrelations(2)and(3)behaveverydifferentlydependingonwhetherthereisanymean?ownormaltothePML.Whenthemean?owisparalleltothelayer,i.e.,Mx=0,thesolutionsarestable.Thisiseasytoseefrom(3)forthevorticitywave.Physically,ifthemean?owisparalleltothePML,thevorticitywavesinthecomputationdomain,beingconvectedbythemean?ow,cannotenterthelayerandhencewouldnotleadtounstablesolution.

ToshowthatforMx=0allthesolutionsof(2)arestable,asimplemappingwillsuf?ce.Rewrite(2)intheform

α2ω2=β2.F≡(ω?βMy)?22(4)

Figure2showstheimageoftheupper-halfω-planeintheFplane.Theupper-halfω-planeismappedintotheentireFplaneexceptfortheslitADC.Butsinceβ2isrealandpositive,

PMLFORLINEARIZEDEULEREQUATIONS

217

FIG.2.Theimageoftheupperhalfω-planeintheF-plane.

forsubsonicmean?owthepointβ2liesoutsidetheimage.Thusnovalueofωintheupper-halfω-planewouldsatisfyEq.(2)indicatingthatthereisnounstablesolution.

2.2.UnstableSolutionsofthePMLEquations

ForMx=0,thePMLequationssupportunstablesolutions.Itistobenotedthat,unliketheoriginaldispersionrelationoftheacousticwaves,Eq.(2)isaquadricequationinω.Ithastwoextrarootsinadditiontothetwomodi?edacousticmodes.Forsmallσ,thetwospuriousrootsaredampedbutoneofthemodi?edacousticrootsisunstable.Forlargerσ,numericalsolutionsindicatethatoneofthespuriousrootsbecomesunstable.Inanycase,theequationsplittingprocedureandtheadditionofanabsorptionterm,botharevitaltothesuppressionofre?ectionsattheinterfacebetweenthecomputationdomainandthePML,inadvertently,leadtoinstabilities.

Forsmallσ,therootsof(2)and(3)canbefoundbyperturbation.Let

(a)(a)(a)+σω1+σ2ω2+···ω(a)=ω0

(v)(v)(v)+σω1+σ2ω2+···,ω(v)=ω0(5)(6)

wheretherootsof(2)and(3)aredesignatedbyasuperscripta(foracousticwaves)andv