2007-0381

45thAIAAAerospaceScienceMeetingandExhibit,8-11January2007,Reno,Nevada

Veri edComputationsofLaminarPremixedFlames

AshrafN.Al-Khateeb ,JosephM.Powers ,andSamuelPaolucci

UniversityofNotreDame,NotreDame,Indiana,46556-5637,USA

Therequiredspatialdiscretizationtocapturealldetailedcontinuumphysicsinthere-

actionzoneforone-dimensionalsteadylaminarpremixedhydrogen-air amesdescribed

bydetailedkineticsandmulti-componenttransportisaccuratelyestimatedaprioribya

simplemeanfreepathcalculation.Toverifythis,arobustmethodhasbeendevelopedto

rigorouslycalculatethe nestlengthscaleaposteriori.Themethodrevealsthatthe nest

lengthscaleisatthemicron-level.Thisresultisconsistentwithanestimatefromtheun-

derlyingmolecularcollisiontheory,andordersofmagnitudesmallerthanthediscretization

scalesemployedinnearlyallmulti-dimensionaland/orunsteadylaminarpremixed ame

simulationsintheliterature.

I.Introduction

Itiswellunderstoodthatinanymathematicallybasedscienti ctheory,associatedcomputationsshouldhave delitywiththeunderlyingmathematics,andtheunderlyingmathematicalmodelhastorepresenttheobservedphysics.The rstissueisdemonstratedbycomparingcomputationalresultswithanotherknownsolutionand/orperformingaformalgridconvergencestudy,whilethesecondissueisdemonstratedbycomparingthecomputationalpredictionswithexperimentaldata.Addressingthesetwoissues,inthisorder,isanecessityinanycomputationalstudytobuildcon denceinboththesimulationstrategyandtheunderlyingmathematicalmodel.

Theexerciseofdemonstratingtheharmonyofthediscretesolutionwiththefoundationalmathematicsisknownasveri cation.1Formulti-scaleproblems,veri cationisdi cultduetotherangeofthespatio-temporalscales,whichmayspanmanyordersofmagnitude.Inthiskindofproblem,usuallymodeledbyhighlynonlinearequations,signi cantcouplingacrossthescalescanoccur,sothaterrorsatsmallscalescanrapidlycascadetothelargescales.Moreover,thestrengthofthecouplingacrossthescalesisnotknownapriori.So,allthephysicalscalesofthemathematicalmodel,temporalandspatial,havetobecapturedinordertohavefullcon dencethatpredictionsarerepeatable,grid-independent,andthusveri able.Subsequently,inthevalidationsteponecanchoosewhatphysicalphenomenaandtowhataccuracyonewantstoreproduceexperiments.

Themainaimofthispaperistorigorouslydeterminetherequiredspatialresolutiontocaptureallphysicalscalesinastandardmulti-scaleproblem:thesteadyone-dimensionallaminarpremixed amepropagatingfreelyatatmosphericpressureinastoichiometricmixtureofhydrogen-airdescribedbydetailedkineticsandmulti-componenttransport.Here,therobustmethodtocalculatethelengthscalesemployedinPowersandPaolucci2,3forgasphasedetonationisimplementedwithmodi cationforde agration.Themethodisrobustinthatithaslittledependenceonthedetailsoftheunderlyingnumericalmethodusedtocalculatethelaminar ame.Itsimplyrequiresalocaldeterminationofthestateofthesystem,whichisfollowedbyaJacobianformulation,andageneralizedeigenvalueanalysis.Assuch,itisabletoestimatewithgreataccuracythelengthscalesonafundamentalmathematical,non-numerical,basis.Theminimumlengthscalewhichmustberesolvedinorderforthemathematicalmodeltobeveri edisthusdetermined.

Inthe rstsection,thegoverningpartialdi erentialequations(PDEs)forunsteadyreactive owarepresented.ThisisfollowedbyareductionofthePDEsintoasystemofdi erentialalgebraicequations(DAEs)

Candidate,DepartmentofAerospaceandMechanicalEngineering,AIAAStudentMember,aalkhate@nd.edu.

Professor,DepartmentofAerospaceandMechanicalEngineering,AIAAAssociateFellow,powers@nd.edu. Professor,DepartmentofAerospaceandMechanicalEngineering,AIAAMember,paolucci@nd.edu.

c2007byJosephM.Powers.PublishedbytheAmericanInstituteofAeronauticsandAstronautics,Inc.withCopyright

permission. Associate Ph.D.

whichdescribesthespatialevolutionofthestatevariables.Followingashortdescriptionofthegeneralizedeigenvalueanalysisandlengthscaledetermination,thestandardformoftheequationsisdelineated,andabriefdescriptionofthenumericalmethodispresented.Next,thenumericalalgorithmisveri edagainstcalculationsgivenbySmookeetal.4Then,themathematicalmodelisvalidatedagainstexperimentaldatacompiledbyDixon-Lewis.5

Forthemainresultsofthestudy,itisdesirabletohaveaphysicalsolutioninallregionsofthelaminar ame.So,inordertosuppressnumericalanomaliesnearthecoldboundarysoastofullyexposethebehaviorinallregionsofthe ame,theinitialmixturetemperatureisraisedto800K.Asaresult,afullyresolvedpredictionofalaminarpremixed ameinastoichiometrichydrogen-airmixtureinitiallyatatmosphericpressureisachieved.Then,allthelengthscalesoverwhichthesystemevolvesareshown,andthe nestlengthscaleiscomparedtopredictionsofasimplemoleculartheory.Moreover,thiscomparisonispresentedforawiderangeofpressures.Finally,acomparisonbetweenthegridresolutionutilizedinmoregeneralrecentstudieswiththerequiredlengthscaletoresolvetheunderlyingone-dimensionalsteadylaminar amestructure,predictedbytheeigenvalueanalysis,isgivenbeforespeci cconclusionsarestated.

II.

II.A.GoverningEquationsMathematicalModel

Thefollowingunsteadyequations6describethesystemunderconsideration,aone-dimensionaladiabaticlaminarpremixedmixtureofNmolecularspeciescomposedofLatomicelementswhichundergoJreversiblereactionswithnobodyforcepresent: (ρu ), x 2 ρu +p τ,(ρu )= x t2 u u 2 pτqρe+= ρu e++J,+ 2 x 2ρρ t

˙iMi,i=1,...,N 1. Yi+Jim)+ω(ρYi)= (ρu x t ρ t= (1)(2)(3)(4)

.ThedependentvariablesareTheindependentvariablesarethespatialcoordinatex andthetimet

mixturedensityρ,mixturevelocityu ,pressurep,viscousstressτ,mass-basedspeci cinternalenergyofqthemixturee,totalheat uxJ,andfortheithspecie,Yi,Jim,andω˙i,whicharethethemassfraction,thedi usivemass ux,andthemolarproductionrateperunitvolume,respectively.TheparameterMiisthemolecularmassofspeciei.Equations(1-3)describetheconservationofmass,linearmomentum,andenergy,respectively.Equation(4)isanevolutionequationforN 1species.

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