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.

Forthissystem,theconstitutiverelationsfordi usivemass uxesandheat uxare

Jim= N MiDikYkMk1 p1 χkT1 Tρ+1 , DiMχk x Mp x T x k=1k=i

N i=1,...,N 1,(5)

Jq

where,= NT D χ1M p1iiiJimhi Tq+,+1 Mχ x Mp x iii=1i=1

T, x

N ρYi

i=1(6)qp== k T(7).(8)Mi

InEqs.(5-8),thenewdependentvariablesaremixture-averagemolecularmassM,temperatureT,Fourier’sheat uxq,andfortheithspecie,χiandhi,whicharethemolefractionandthemass-basedspe-

Tci centhalpy.ThevariablesDik,k,andDiarethemulti-componentdi usioncoe cients,thetemperature-

dependentmixturethermalconductivity,andthethermaldi usioncoe cientofspeciei.Aconstantpa-rameteristheuniversalgasconstant =8.314×107erg/mole/K.Equations(5-6)areappropriateforamixtureofidealgases,7anddescribemulti-componentmassdi usion uxesincludingtheSorete ect,andtheheat uxincludingtheDufoure ect.Equation(7)de nesFourier’slaw.Equation(8)isthethermalstateequationforanidealgasmixture.Thissystemofequationsiscompletedbyadoptinganappropri-atesetofadditionalconstitutiverelations,(e.g.thelawofmassaction,theArrheniusreactionrate,thetemperature-dependententhalpy,andentropy).FulldetailsaregivenbyAl-Khateebetal.8andSinghetal.9

Thecompletesystemissimpli edbyreducingitintoasystemofordinarydi erentialequations(ODEs).Thetime-dependentbehaviorofthesystemisrelaxedtoasteadilypropagating amefrontwithconstant,albeitunknown, amespeedS.Thelow-Machnumberassumptionisadopted.Thisassumptionisreason-ableforde agration,10andimpliesthatfora xedmass uxthemomentumequationnolongerneedbeconsidered.AsystematicreductionofthecompletesystemisgiveninRef.8.

Asaresult,thegoverningequationsarerecastintheformd(ρuh+Jq)d(ρuYle+Jle)dxd(ρuYi+Jim)dxd(ρu)dx===0,0,0,l=1,...,L 1,i=1,...,N L.(9)(10)(11)(12)=ω˙iMi,

Thespatialcoordinatexisa amefront-attachedcoordinate,uisthemixturevelocityinthe ameframe,histhemass-basedspeci centhalpyofthemixture,Ylearetheelementmassfractions,andJlearetheelementmass uxes.ForEqs.(9-12),thatdescribethesteadilypropagatinglaminarpremixed ame,theappropriatesetofboundaryconditionsis

x=0:

∞:xf:Jim=Yio,i=1,...,N 1,T=To,Yi+odTdYi→0,→0,i=1,...,N 1,dxdxT=Tf,(13)(14)(15)x→x=

wherexfisaspeci edspatialpointandTfisthespeci edtemperatureatthatlocation.11Thesearecommonlyusedtostudyde agration,thoughotherformulationsarepossible.

AtthisstagethevariableSisconsidereda xedparameterforagivencalculation;aniterativetechniqueisusedtodetermineSsothatallboundaryconditionsaresatis ed.Theequationsaremostconvenientlyposedasasetof2N+2DAEsintermsof2N+2statevariables;speciesmassfractionYi,(i=1,...,N),speciesmass uxJim,(i=1,...,N),temperatureT,andFourier’sheat uxq.Thissystem,inacompactrepresentation,is

A·dz=f,dx(16)

D ˙A= M0 0I0 0 0 ,z= Q

where,

D

˙M

I

Q==== . .. Y1m JNM .N .. ω˙1M1 YN . .m . J1 ,f= ω˙N LMN L.. . ρuYe+Je (ρoSYe) 111o m JN .. . T eee+JL (ρoSYLo) ρuYLq q ρuh+J (ρoSho)q mJM1 , (17)N DimYmM,Dik Mkm=1i=1,...,N,0(N L)×L0L×L ,k=1,...,N,(18)I(N L)×(N L)0L×(N L) 00. k0 m=iρoSI(N L)×(N L)0L×(N L),(19)(20)(21)0(N L)×L0L×L

Thedynamicalsystem,Eq.(16),andtheboundaryconditions,Eqs.(13-15),areusefulforlengthscaleanalysis.Directsolutionofthissystemforthereactionzonestructureispossible,inprinciple.However,theproblemcanbeshowntobeahighordershootingproblemrenderingdirectsolutiondi cult.

II.B.APosterioriLengthScaleAnalysis

Toaccuratelydeterminethelengthscalesoverwhichthesystemevolves,aneigenvalueanalysiscanbeappliedtoEq.(16).SinceAissingular,thestandardeigenvalueanalysisisnotapplicable.Instead,thegeneralizedeigenvaluesofthisdynamicalsystemcanbecalculated.12

Assume rstthatz= z(x)hasbeendeterminedbysomeappropriatenumericalmethodsothat z(x)satis esEqs.(13-15,16).Considerthenanarbitraryspatialpointx=x atwhichthestatevariablesarez= z(x )=z .Byde ningtheperturbationfrom z(x)as z(x)=z(x) z(x),linearizingEq.(16)aboutx=x ,andadoptingthestandardassumptionthat

z=veλx,(22)

whereλandvareconstantsyieldthegeneralizedeigenvalueproblem

λA ·v=B ·v,(23)

whereλisingeneralacomplexnumberdenotingthegeneralizedeigenvalue,visthecorrespondinggener-alizedeigenvector,andA andB arematricesofsize(2N+2)×(2N+2).FulldetailsaregiveninRef.8.Solvingforλi,i=1,...,2N L,thenusingEq.(22),itiseasilyseenthatthelengthscalesoverwhichthedependentvariablesevolvearegivenbythereciprocaloftherealpartoftheseeigenvalues,

i=1,|Re(λi)|i=1,...,2N L.(24)

Byevaluatingtheeigenvaluesateachspatialpoint,thelengthscalesoverwhichthesystemevolvesthroughthereactionzonearedetermined.Asaresult,theminimumsizeofdiscretizationtocapturethe

nestscaleofthesystemcanbedetermined.Ingeneral,theeigenvaluesarecomplex,wherethereciprocalsoftherealpartsprovidethelengthscalesofamplitudegrowth,andthereciprocalsoftheimaginarypartsrepresenttheoscillatorylengthscale.Inthiswork,theeigenvaluesarepurelyreal,exceptforsomelimitedregions.

II.C.StandardFormoftheEquations

Thelessre ned,butmorecompact,formwhichcommonlyappearsintheliteraturetomodelstationarylaminarpremixed ames,cf.Refs.11,13,14,areobtainedfromEqs.(1-4)byfollowingthesameapproachasAris15toarriveatd(ρu)=0,dx N dhdq dTiJim+++ω˙iMihi=0,ρucpdxdxi=1dx

dYidJimρu+dxdx=ω˙iMi,i=1,...,N 1.(25)(26)(27)

Asolutionforthisboundaryvalueproblem,Eqs.(25-27)withtheboundaryconditionsEqs.(13-15),canbeobtainedbydiscretizingthespatialdomainusing nitedi erences,andtheresultingalgebraicsystemofequationsaresolvediterativelyusingadampedmodi edNewton’smethod,wherethesolutioniterateisbroughtintotheconvergencedomainbyusingpseudo-timeintegration.16

http://putationalMethod

AdoubleprecisionFORTRAN-77codehasbeendevelopedandlinkedwiththeInternationalMathematicalandStatisticalLibraries(IMSL)routinesDFDJACforJacobianevaluation,DEVLRGforeigenvaluesestimation,DGVLRGforgeneralizedeigenvaluesestimation,andadoubleprecisionversionofthepublicdomaineditionoftheCHEMKINpackage17,18toobtainkineticratesandthermodynamicsproperties,adoubleprecisionversionofthepublicdomaineditionoftheTRANSPORTpackage19tocalculatemulti-componenttransportpropertiesofspecies,andadoubleprecisionversionofthepublicdomaineditionofthePREMIXalgorithm16toobtainthesteadystructureofadiabaticlaminarpremixed ames.

Inthisstudy,theresolvedstructureisobtainedbysolvingthestandardform,Eqs.(25-27),andtheeigenvaluesarecalculatedbyusingthedynamicalsystemform,Eq.(16).Moreover,allresultsareobtainedonagridthathasbeenadaptivelyre nedtocaptureregionsofsteepgradient.Asecondordercentraldi erenceschemehasbeenemployedtodiscretizeallthespatialderivatives.Themassandheat uxeshavebeenestimatedatintermediategridpointstomaintainsecondorderaccuracy.Moreover,allthecalculationspresentedinthisworkwereperformedonasingleprocessor3.2GHzHewlett-Packardmachine,andtypicalcalculationsfortheeigenvalueswerecompletedwithinoneminute.

IV.Results

Astoichiometrichydrogen-airmixtureatpo=1atmhasbeenconsidered,wheretheinitialmolarratioisgivenby2H2+O2+3.76N2.AkineticmodelidenticaltothatofSmookeetal.,4withL=3elements,N=9species,andJ=19reversiblereactionsisused.Inthismechanism,thereactantspeciesareH2,O2,H,O,OH,HO2,H2O2,andH2O.TheinertdiluentforthemixtureisN2.

IV.A.MathematicalVeri cationandExperimentalValidation

Anecessaryveri cationistoachievethesameresultsthathavebeenobtainedinpreviousstudies.Tothisend,acalculationisperformedtoreproducethetemperatureandspeciespro lesofastoichiometric,atmosphericpressurehydrogen-air amefoundinSmookeetal.4Equations(25-27)withtheboundaryconditionsEqs.(13-15)aresolved,wherethismathematicalmodelisidenticaltotheonedescribedinRef.4.Thespeci edspatialpointisxf=0.05cm,thespeci edtemperatureisTf=400K,andthetemperatureoftheunburnedmixtureisTo=298K.Inthisparticularcalculation,theDufoure ectintheheat uxmodelisneglected,whiletheSorete ectinthemass uxmodelisconsideredtomatchthemodelinRef.4.

AlthoughconsideringoneofthesetermsandneglectingtheotherviolatesOnsagerreciprocity,thisisdonehereforveri cationpurposesonly.TheresultsareillustratedinFig.1;itiseasilycheckedthatthestationary amestructureisidenticaltothatofRef.4.

Next,acomparisonwithexperimentalresultsaddressesthequestionastowhetherthemodelisagoodrepresentationoftheobservablephysics.Forvalidationpurposes,aseriesofcalculationsisperformedonanatmosphericpressurehydrogen-airlaminarpremixed ameinitiallyatTo=298K.Atdi erentH2percentagesinthemixture,the amespeedisdetermined.Acomparisonbetweenthecalculated amespeedsandthedatacompiledbyDixon-Lewis20ispresentedinFig.2.Itisclearthatthecomputationalpredictionsliewithinthescatteroftheexperimentaldata,andtheyareasgoodashavefoundbyothers.4,20IV.B.StoichiometricHydrogen-AirPremixedLaminarFlame

Next,thePREMIXcode16isusedfordeterminingthestationarystructureofaone-dimensional,stoichiometric,adiabatic,2H2+O2+3.76N2premixedlaminar ameatatmosphericpressure.Thespeci edtemperatureisTf=900K,thespeci edtemperaturelocationisassignedatxf=2.30cm,andthemixturetemperatureatthecoldboundaryisTo=800K.

Fullyresolvedsteadytemperatureandspeciespro lesforTo=800KareshowninFigs.3and4.Althoughlinearscalesareusuallyusedintheliterature,herelog-logandsemi-logscaleshavebeenemployedtobetterillustratethedisparatescales.Figure3showsthespatialdistributionofspeciesmassfractionsthroughouttheentire amezone.Atx≈10 4cm,theminorspeciesgrowthrateschangeslightly,whichrevealsthatsigni cantdissociationreactionsatthisscaleareinduced.Anotherincreaseintheminorspeciesmassfractiongrowthratesisnotedatx≈10 2cm,whichindicatestheoccurrenceofmorevigorouschemicalinteractionoftheminorspecies.For10 2<x<2.30×100cm,theminorspeciesmassfractionscontinuetoincreaserapidlywithdi erentgrowthrates.Ontheotherhand,themajorspeciesH2,O2,andN2haveessentiallyconstantmassfractions.Justpastx=2.20×100cm,whichisneartheendofthepreheatzone,allthespeciesmassfractionsundergosigni cantchange,andtheradicals’massfractionsreachtheirmaximumvalues.Atx=2.40×100cm,exothermicrecombinationofradicalscommencesformingthepredominantproductH2O.Thiszoneextendsuptox=1.39×101cm;afterthat,thesystemcomestoanequilibriumwhereallthespatialgradientsvanish.Tocon rmthis,thespatialdomainwasextendedtox=1.00×102cm,butnofurtherchangeswerenoted.

InFig.4,thetemperaturepro leispresented.Atx≈2.20×100cm,thereactionundergoesaparticularlyvigorousstageinwhichthechangeinthetemperatureissigni cant.Thus,theignitionpointcanbeassigned;itisde nedasthepointwherethetemperaturegradientdT/dxreachesamaximumvalue.Also,thisparticularpointde nestheendofthepreheatzoneandthebeginningofthereactionzone,whichextendstotheequilibrium.Forthiscasetheignitionpointisassignedatx=2.315×100cm,andthereactionzonelength,de nedasthedistancefromtheignitionpointuptothelocationwherethetemperaturereaches0.99ofitsequilibriumvalue,is reaction=1.16×101cm.

Havingthefullyresolvedstructureinhand,thelocalJacobianandthespatialgeneralizedeigenvaluesarecalculatedfromthecoldboundarytonearequilibrium.Asaresult,thelocallengthscales iarepredictedthroughoutthedomain,Fig.5.Themulti-scalenatureoftheproblemandthelengthscalesoverwhichthespeciesevolveareclearlyshown.The nestlengthscaleandthelargestlengthscaleforthissystemvaryfrom7.60×10 4cmand1.62×107cminthepreheatzoneto2.41×10 4cmand2.62×100cminthereactionzone,respectively.

InadditiontothevaluesreportedinFig.5,theeigenvalueswerecheckedbycalculationwithotherstandardalgorithms.Allalgorithmsreturnedequivalenteigenvaluescorrespondingto nelengthscales.However,numericalerrorsinducedsomediscrepanciesinthelessimportantcoarselengthscaleestimates.Theleastnumericalerrorinthephysicaleigenvalueswasnotedinthedirectcalculationoftheeigenvalues;however,thisalgorithmreturnedL=3spuriouseigenvalues,whichwereoverruledbythemorerobustgeneralizedeigenvaluemethod.

Theevolutionofaparticularspeciesisnotassociatedwithaparticularlengthscale,sincethespeciesmassfractionsdependonlocallinearcombinationsofalleigenmodes.So,thespeciesmassfractionsvaryonthesescalesthroughtheentiredomain.Theimportant nestscaleis finest=2.41×10 4cm,whichoccursatequilibrium.Thepredicted nestlengthscaleandthesmallestscaleoverwhichthespeciesvary,x=10 4cm,arenearlyidentical.Moreover,the nestlengthscalee ectinthepreheatzonecanbeobservedinthevariationoftheminorspeciesmassfractions,whichensurestheconsistencybetweentheeigenvalue-determined nestlengthscaleandthesmallestscaleoverwhichthespeciesvary.Asthesystem

approachesequilibrium,alloftheeigenvaluesarereal:halfarepositive,andhalfarenegative.Thus,theequilibriumpointisahighordersaddlenode.

IV.C.MeanFreePathEstimate

Remarkably,itispossibletodevelopasimpleaprioriestimatefor finestpredictedaposterioribythecalculationsoftheprevioussection.Suchanestimateisusefulinprovidingalowerboundforcomputationalgridresolutionnecessarytoguaranteefullyresolvedcontinuumcalculations.The nestlengthscalecanbeshowntobeclosetothemixturemeanfreepathscale mfp,whichisthecuto lengthscaleassociatedwithcontinuumtheories.Physicallymeaningfulresultsareavailableatorabovethemeanfreepath.AsimpleestimateforthemeanfreepathisgivenbyVincentiandKruger,21

mfp=M.Nπd2ρ(28)

Here,thenewparametersaredthemolecularcollisioncross-sectiondiameterandN=6.0225×10mole 1Avogadro’snumber.Forthecalculationof mfp,theestimateofd=3.70×10 8cmforairisadoptedfromRef.21;M=28.00g/moleandρ=1.11557×10 4g/cm3wereeasilyquanti edforthemixturefromthePREMIXcalculations.Theestimaterevealsthat mfp=5.87×10 5cm,whichisroughlyoneorderofmagnitudesmallerthanthecontinuum-based finest.

Followingthesameprocedure,acomparisonbetweenthepredicted nestlengthscale finest,themeanfreepathlengthscale mfp,andthereactionzonelength reactionoverawiderangeofpressuresispresentedinFig.6.Itrevealsthattheestimated nestlengthscaleiswellcorrelatedwiththemeanfreepathandthatboth mfpand finestdecreasesimilarlyaspressureisincreased.Thisisconsistentwiththefactthattheparametersusedintheconstitutivemodelsforthecontinuumtheoryhaveastheirfoundationtheaveragednatureofthemorefundamentalcollisiontheory.Inaddition,both mfpand finestareatleastthreeordersofmagnitudelessthanthereactionzonelength reaction.

Thisapproachhasbeenextendedtoseveralhydrocarbonmixtures(methane,ethane,propane,ethylene,andacetylene),awiderangeoffuel-airratios,andanotherreactive owcase,theChapman-Jouguetdet-onation.Severaldetailedkineticsmodelshavebeenadopted(e.g.GRI-3.0,GRI-1.2).TheresultsoftheseextensionsaregivenindetailinRef.8.Itwasfoundthattheapriorimeanfreepathestimatepredictedthe nestlengthscaleaccuratelyforallcases. 23

http://parisonwithPreviouslyPublishedResults

Thoughthisworkfocusesonestimatingthelengthscalesofone-dimensionalsteadylaminarpremixed ames,theestimatesprovideboundsforotherproblemswheremulti-dimensionalandunsteadye ectsaresimulated.Inthissection,acomparisonbetweenthepredicted nestlengthscaleandtheutilizeddiscretizationinsomeofthebestcalculationsoflaminarpremixed amesinhydrogen-airmixturesispresented.TheresultsofthesecalculationsaresummarizedinTable1,whichisorganizedsuchthatforeachstudytheinitialmixturemolarratio,temperature,pressure,reactionzonelength reaction,estimatedmeanfreepathlength mfp, nestlengthscale finestpredictedbythegeneralizedeigenvalueanalysis,andgridresolution xemployedarelisted,respectively.Inallcases,thepredicted nestlengthscalesareatthemicron-level,andtheyarewellcorrelatedwiththeassociatedcuto lengthscalesadmittedbythecontinuumtheory.

KattaandRoquemore22investigatedthestructureofanaxi-symmetricpremixedhydrogen-airjet ameusingatime-dependenttwo-dimensionalalgorithm.Theutilizeddiscretizationwasnonuniform,andtheminimumgridsizeintheaxialdirectionwas2.50×10 2cm.ThedetailedkineticsmodelconsistedofN=11speciesandJ=20reversiblereactions.Ithasatypographicalerrorinreaction5 6whichisunbalanced.ThiserroriscorrectedbyreturningtotheworkofWestbrook23andadoptingthecorrespondingreactions.Thieleetal.24usedatime-dependenttwo-dimensionalmodeltosimulatethespark-ignitioninaquiescenthydrogen-airmixturedescribedbyadetailedkinetics.ThereactionmechanismconsistsofN=9speciesandJ=38irreversiblereactions,adoptedfromtheworkofWarnatzetal.25Althoughthegriddiscretizationinthisstudyisnotmentioned,thepredictedrequiredlengthscale,7.56×10 4cm,tofullyresolvesuchasystemwasbeyondtheexistingcomputationalcapabilities.

PatnaikandKailasanath26usedadetailedkineticsmodelextractedfromtheworkofBurksandOran27tosimulateatwo-dimensionalburner-stabilizedhydrogen-air ame.TheextractedmodelconsistsofJ=48elementaryreactionsinvolvingN=9species,buttheoriginalmodelhastypographicalerrorsinreactions

4and12,whichareunbalanced.TheseerrorsarecorrectedbyreturningtotheworkofBaulchetal.28andHampsonetal.29andadoptingthecorrespondingreactions.Thespatialresolutioninthisstudywasnonuniform,thoughtheaveragegridsizewas x=3.54×10 2cm.

ThemainresultfromTable1isthatnoneofthesestudieshaveutilizedagridresolution xthatislessthanorequaltothe nestlengthscale finestwhichisrequiredtohaveunambiguouslyresolvedresultsforasteadyone-dimensionallaminarpremixed ameincomparablemixtureunderthesameconditions.Moreover,theutilizedgridresolution xisatleasttwoordersofmagnitudegreaterthan finest.Ineachstudy,di erentphysicalphenomenaaresimulated,andthemathematicalmodelsthatareusedvary,butthecommonalityinallstudiesistheusageofadetailedkineticsmodeltosimulates ameinapremixedhydrogen-airmixture.

Lastly,ourresultsareinroughagreementwithindependentestimatesfoundindirectnumericalsimulation(DNS)ofturbulentreacting ows.Inarecentstudy,Chenetal.,30presentatwo-dimensionalDNSofautoignitionatconstantvolumeandhighpressureofhydrogen-airdescribedbydetailedkinetics.Thedomainsizewas4.1mm×4.1mm,andthecalculationsrequiredagridresolutionof x=4.30×10 4cmtoresolvetheignitionfronts,whichissimilarinmagnitudetothe nestlengthscalepredictedhere.

V.Conclusion

Thepresentone-dimensionalsteadycalculationsrevealthatforanadiabaticlaminarpremixed amefreelypropagatinginstoichiometricmixturesofhydrogen-airdescribedbydetailedkineticsandmulti-componenttransport,therequiredgridresolutiontoformallyresolvethemodeled owstructuresisatthemicron-level.Thislengthscalehasbeenpredictedbyutilizingarigorouseigenvalueanalysis.Thelengthscalepredictionsarefullyre ectiveoftheunderlyingphysicsandnottheparticularnumericalmethodchosen.Thishasbeenveri edbyshowingthatthe nestlengthscaleiswell-correlatedwiththemeanfreepathcuto lengthscaleestimatedfromkinetictheory.Thus,itispossibletouseasimplemeanfreepathcalculationasanaprioriestimateofthelowerboundforgriddiscretization.Relatedcalculationsofunsteadyandmulti-dimensionallaminar amesintheliteraturetypicallyemploymuchlargerdiscretizationsthansuggestedbythepresentanalysis.Thefullconsequencesofthisunder-resolutionawaitrigorouslinearandnon-linearstabilityanalysisaswellasDNSinordertobedetermined.

Acknowledgments

TheauthorsrecognizethesupportoftheChemistryDivisionofArgonneNationalLaboratoryandhelpfulconversationswithDr.MichaelJ.DavisofArgonne.The rstauthoracknowledgesthepartialsupportofthisworkfromtheCenterforAppliedMathematicsatUniversityofNotreDame.

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http://parisonoflengthscalesamongvariousmodelsthatusedetailedkineticstodescribealaminarpremixedhydrogen-air ame.Ref.

22

24

26Mixturemolarratio1.26H2+O2+3.76N21.19H2+O2+3.76N2

0.59H2+O2+3.76N2To,(K)4.00×1023.05×1023.50×102po,(atm)1.000×1000.987×1001.000×100 reaction,(cm)1.52×1011.31×1011.49×101 mfp,(cm)4.33×10 53.99×10 57.84×10 6 finest,(cm)8.05×10 47.56×10 44.35×10 5 x,(cm)2.50×10 2—3.54×10 2

x[cm]

Figure1.Temperatureandspeciespro lesvs.distanceinastoichiometrichydrogen-air amefornumericalveri cation,equivalenttopredictionsofSmookeetal.,4To=298K,po=1atm.

T

[K]χi

S[cm/sec]χH2

http://parisonofpredictionsof amespeedvs.theunburneddiatomichydrogenmolefractionwiththeexperimentaldatacompiledbyDixon-Lewis,20To=298K,po=1atm.

10

10 5Yi

101010

10101010101010x[cm]

Figure3.Speciesmassfractionvs.distanceforastoichiometrichydrogen-air ame,To=800K,po=1atm.

Figure4.Temperaturevs.distanceforthestoichiometrichydrogen-air ame,To=800K,po=1atm.

108

10

6

104

[cm]102

i

100

10 2

10 4

10 510 410 310 210 1100101102

x[cm]

Figure5.Predictedlengthscalesoverwhichstoichiometrichydrogen-air ameevolvevs.distance,To=800K,po=1atm.

Figure6.Thereactionzonelength,the nestlengthscalepredictedbyeigenvalueanalysis,andthemeanfreepathvs.pressureforstoichiometrichydrogen-air ame,To=800K.