Flame–wall interaction effects on the flame root stabilization mechanisms of a doubly-transcritical LO2/LCH4 cryogenic flame

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ersion :

Laurent, Charlélie and Esclapez, Lucas and Maestro, Dario and

Staffelbach, Gabriel and Cuenot, Bénédicte and Selle, Laurent and

Schmitt, Thomas and Duchaine, Florent and Poinsot, Thierry Flame–

wall interaction effects on the flame root stabilization mechanisms of a

doubly-transcritical LO2/LCH4 cryogenic flame. (2019) Proceedings of

the Combustion Institute, 37 (4). 5147-5154. ISSN 1540-7489

O

pen

A

rchive

T

oulouse

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rchive

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uverte (

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Flame–wall

interaction

effects

on

the

flame

root

stabilization

mechanisms

of

a

doubly-transcritical

LO

2

/LCH

4

cryogenic

flame

C. Laurent

,

L.

Esclapez

,

D.

Maestro

,

G.

Staffelbach

,

B.

Cuenot

,

L. Selle

,

T.

Schmitt

,

F.

Duchaine

,

T.

Poinsot

a_{CERFACS,42AvenueGaspardCoriolis,ToulouseCedex131057,France} b_{IMFT,AlléeduProfesseurCamilleSoula,Toulouse31400,France}

c_{LaboratoireEM2C,CNRS,CentraleSupélec,Université Paris-Saclay,3,rueJoliotCurie,91192Gif-sur-Yvettecedex,} France

Abstract

High-fidelitynumericalsimulationsareusedtostudyflamerootstabilizationmechanismsof cryogenicflames,wherebothreactants(O2 andCH4)areinjectedintranscriticalconditionsinthe geometryof thelaboratoryscaletestrigMascotteoperatedbyONERA(France).Simulationsprovidea detailedinsightintoflamerootstabilizationmechanismsforthesediffusionflames:theyshowthatthelarge wallheatlossesatthelipsofthecoaxialinjectorareofprimaryimportance,andrequiretosolveforthefully coupledconjugateheattransferproblem.Inordertoaccountforflame–wallinteraction(FWI)atthe injectorlip,detailedchemistryeffectsarealsoprevalentandadetailedkineticmechanismforCH4 oxycombustionathighpressureisderivedandvalidated.Thiskineticschemeisusedinareal-gasfluid solver,coupledwithasolidthermalsolverinthesplitterplatetocalculatetheunsteadytemperaturefield inthelip.Asimulationwithadiabaticboundaryconditions,anhypothesisthatisoftenusedinreal-gas combustion,isalsoperformedforcomparison.Itisfoundthatadiabaticwallssimulationsleadto enhancedcryogenicreactantsvaporizationandmixing,andtoaquasi-steadyflame,whichanchorswithin theoxidizerstream.Ontheotherhand,FWIsimulationsproduceself-sustainedoscillationsofbothlip temperatureandflamerootlocationatsimilarfrequencies:theflamerootmovesfromtheCH4 totheO2 streamsatapproximately450Hz,affectingthewholeflamestructure.

Keywords: Cryogenicflame;Flame–wallinteraction;Conjugateheattransfer;Real-gasthermodynamics;Flame anchoring

∗_{Correspondingauthor.}

E-mailaddress:laurent@cerfacs.fr(C.Laurent).

1. Introduction

Theglobalmarketofspacelauncherswitnessed amarked andfastevolutionatthebeginning of

the2010s,withincreasedeffortstodrasticallycut downoperation costsforexample bydeveloping reusablelaunchers,suchastheFalcon9,designed by SpaceX. These successes were largely due to technologicaladjustmentsonliquidrocketengines (LREs),whicharenow expected tobere-usable, re-ignitable,andabletooperateatvariable-thrust levels.Tomeettheseconstraints,enginedesigners plantoemployCH4/O2cryogeniccombustion,in variousthermodynamicstatesatinjection,namely supercritical conditions (LOx/GCH4) as well as doublytranscriticalconditions(LOx/LCH4).Even thoughthesenewLREsareexpectedtobefully ex-ploitableby2030,littleisknownonthesenew real-gasdiffusionflamesstabilizedoncoaxialO2/CH4 injectors.

Fundamental reviews of physical processes governingmixing, vaporization, and combustion under transcritical and supercritical conditions are proposed in [1,2]. In the transcritical state (i.e.,whenpressureexceedsthecriticalpressureof thefluid,buttemperatureisbelowitscritical tem-perature)thefluidthermodynamicproperties sig-nificantlydifferfromthatofaclassicsubcritical liq-uid:assurfacetensionvanishesjetdynamicsareno longercontrolledbybreakupandatomization,but by mass transfer throughturbulent mixing [3,4]. Significanteffortswerecarriedoutinthedomain of numerical simulation of real-gas combustion withpioneeringworks[5–7] investigatingrealistic LO2/GH2 shear-coaxial jet flames. It was found thattheflameroot,dominatedbydiffusioneffects duetolargethermophysicalpropertiesgradients,is stabilizedinaregionofhighshearbehindthe in-jectorlip.Meanwhile,experimentsofsimilar super-criticalreactiveflowshavebeencarriedoutin dif-ferentfacilities,includingtheMascottetestbench operatedbyONERA(France)[8].Firstthe struc-tureofsupercriticalLO2/GH2shear-coaxialflames wascharacterizedexperimentally[9–11],andthese resultswerelaterrecoverednumericallywithLarge EddySimulation(LES)[12].LO2/CH4flames un-der various injection conditionswere also exam-inedexperimentally[13]andnumerically[14].

Mostpreviousresearcheffortsfocusedon super-criticalflameswhereoxygenwasinjectedinadense transcriticalstate,andfuel(eitherH2orCH4)asa lightsupercriticalfluid(i.e.,atatemperatureabove itscritical temperature).Veryfewstudies tackled theproblemofadoublytranscriticalregime,where bothreactantsareinjectedinadensetranscritical state.TheexperimentalworkofSinglaetal.[13]is oneof theonlyanalysisreporting characteristics of suchflamesinarealisticshear-coaxial config-uration,wheretheflamedisplaysaspecific struc-tureconsistingoftwoconcentricconicallayersof reactions. Asthis combustion regime islikely to occurin futureLREs,a betterunderstanding of governingphenomenainthesespecificconditions ishighly needed. Inaddition,most previous nu-mericalapproachesreliedontwocrude

simplifica-tions:(1)adiabaticboundaryconditionswere as-sumedatthecoaxialinjectorlip[6,7],and(2) sim-plifiedchemistry,tabulated[14,15]orbasedon two-stepreducedschemes[12],wasemployed.However, asbrieflymentionedin [6],heattransfer islikely tobeimportantinthecoaxialinjectorlip,dueto thecombinationofveryhighheatfluxesandthin lips.Thiseffectisexpectedtobe evenmore pro-nouncedinthecaseofdoublytranscritical combus-tion,sincebothreactantsareinjectedinacryogenic state.Asaconsequence theresolutionof afully coupledconjugateheattransferproblemis neces-sarytoaccuratelyaccountfortemperatures varia-tionswithinthewall.

Followingtheseobservations,thepresentwork usesatwo-dimensional quasi-DNS method with complex chemistry to investigate flame root sta-bilization mechanisms for a doubly transcriti-cal LO2/LCH4 combustion in the Mascotte test rig[13].ThispaperaddressesFWIeffectsonflame rootstabilization,byconsideringtheunsteady con-jugateheattransferproblem,wheretheflameand theliptemperaturefieldarestronglycoupled.To theknowledge of the authors,thisisone of the firstattemptstosimulatedoublytranscritical com-bustionofshear-coaxialflamestypicalof modern LRE,includingFWIanddetailedCH4/O2 chem-istry.Section2describesthenumericalapproach, includingdetailsconcerningthecouplingstrategy betweenthefluidsolverandthewallthermalsolver, aswellasabriefsummaryof theoperating con-ditions.InSection3adetailedkineticmechanism forCH4oxycombustionisderivedandvalidatedfor highpressureconditions.Finally,Section4presents ananalysisofflamerootstructureanddynamics, obtainedforafullycoupledsimulationand com-paresthemtoacasewherethelipwallsareassumed adiabatic.

2. Numericalapproachandoperatingconditions Theunstructured solverAVBP [14,16]isused tosolvethetwo-dimensionalcompressibleNavier– Stokesequationsforareactivemulti-components mixture of fluids. Real-gas non-idealitiesare ac-countedforthankstotheSoave–Redlich–Kwong (SRK)equationofstate[17]: p= ρ ¯rT 1− ρbm − ρ2am(T) 1+ρbm (1) where ¯ris themixturegas constant,andbm and

am(T)arecalculatedasin[1].Numericalschemes andboundaryconditionsareadaptedforrealgas thermodynamics. A specific local filtering proce-dureisalsousedtohandlestrongdensityand pres-sure gradients caused by real-gas non-idealities. Moredetailsaboutthisprocedureaswellasother numericalparameterscanbefoundin[14,18]. Ac-counting forFWI in theflame root stabilization

Table1

Injectionconditions(massflowrate,reducedpressuresandtemperatures),andfluidthermalproperties(effusivitye(SI), andeffusivityratioκf)forreactantsandhotgases(HG).Theglobalequivalenceratioisφg=14.3,themomentumflux

ratioisJ=(ρFuF2)/(ρOu2O)=33.3,andthewalleffusivityisew=1.9× 104(SI).

Injectionconditions Thermalproperties ˙ min j/m˙Oin j2 T in j R P in j R ef(SI) κf O2 1.0 0.55 1.49 260 75.5 CH4 3.59 0.62 1.64 2200 8.9 HG – 80 245

Fig. 1. The two-dimensional computational domain. Thickdarklinesrepresentno-slipadiabaticwallsforthe fluiddomain.Dashedlinesarenon-adiabaticno-slipwalls werefluidandsoliddomainsarecoupled.Thebasisofthe lip(hashedline)isanadiabaticboundaryconditionfor thesoliddomain.

mechanisms requires solvinga fullycoupled un-steadyconjugateheattransferproblem[19].Here, athermalsolvernamedAVTPisusedforthe res-olutionof heatconductioninthecoaxialinjector lip. Bothfluid andthermalsolvers arerunusing aParallelCouplingStrategywhereexchange infor-mationisperformedthroughtheOpenPalm super-visor[20,21]:theflowsolverprovidesafluxtothe conduction solver,whilethisoneprovidesa tem-peraturetothefluidsolver.Betweentwocoupling events,theflowsolver(resp.theheatsolver) per-formsαf (resp.αw) iterationswithatimestep dtf (resp.dtw),andexchangefrequenciesareadjusted suchthat:αfdtf=αwdtw,inordertoensure phys-icalsynchronizationandcontinuityof fluxes and temperaturesattheinterface.

TheMascottegeometryconsistsofacoaxial in-jectorwithacentraloxygenstreamofdiameterdO2 surroundedbyanannularmethanestream[13,14]. As the present study focuses on flame root sta-bilization,computationsareperformedona two-dimensionaldomainrefinedinthenear-injector re-gion(Fig.1).Theareaof interest,of dimensions 2dO2× 1dO2,islocatedneartheinjectorlipof thick-nessδlip,andconsistsofcellsofdimension0.03δlip (i.e.,afewμm).Thediffusionflamereactive thick-ness(oforder100μmforthestrainratelevels con-sideredhere)impliesahighresolutionofthefront withatleast10–16cellsacrossthereactive thick-ness. Injectionconditions(Table1)aresimilar to

theoperationpointT1in[13].Theonlydifference is that thechamber pressure has been increased to75 bar (instead of 54 bar intheexperimental study),givinghigher reducedpressures.This dis-crepancywiththeexperimentalconditionsof[13]is expectedtostronglyaffecttheflowfeatures,and di-rectcomparisonwithexperimentaldataof [13]is thereforeruledout.However,theaimofthepresent workisnottoquantitativelyreproduce experimen-talresults,butrathertoqualitativelydescribeflame stabilizationmechanismsoccuringinLREs condi-tions.Thecomputationofthishigherpressure op-erationpointwas motivatedbytwoelements:(1) apressureof75barisstillrelevanttoLREs con-ditions,anditremainsaworthyqualitative analy-sis;(2)attheexperimentalpressure(54bar),both reactantspass neartheir respectivecritical point as theyare heated up, resulting in high fluctua-tionsof pressureanddensity,duetothe numeri-calresolutionofextremelylargedensitygradients combinedwithhighlynonlinearthermodynamics. These numerical oscillationsdestabilize the com-putation,unlessasignificantartificialviscosityis added.Onthecontrary,at75barthesenumerical oscillationsarelessimportantanddonotrequire the addition of an important artificial viscosity, thusyieldingahigher-fidelitycomputation. One-seventhpower-lawvelocityprofilesareimposedat both reactants inlets throughNSCBC boundary conditions[22]adaptedforreal-gas thermodynam-ics.Onlythetemperaturefieldinthelipofthe cen-tralinjectoriscoupledwiththefluiddomain.The meshusedbythethermalsolverisuniform,witha finercellsizeofδlip/120,inordertoavoidaliasing duetointerpolationerrorsattheinterface.Atthis pointitisinterestingtodefinesomequantities rel-evanttoconjugateheattransfer.Thefluid effusiv-ityisdefinedasef =(ρCpλ)1/2,withλ thethermal conductivityandCpthespecificheatcapacity.We alsodefinethefluideffusivityratiobyκf =ew/ef. The effusivity allowsto evaluatetheability of a bodytoexchangeheatwithitssurrounding.Thus forκf 1,temperatureatthesolid/fluidinterface ismostlydeterminedbythewalltemperature,and thisonecanbeconsideredasisothermal.Onthe contrary,ifκf 1theinterfacetemperatureis im-posedbythefluidandthewallmaybeconsidered asadiabatic.Intermediatevaluesofκfcorrespond toboundariesthatareneitheradiabaticnor

isother-Fig.2.ComparisonbetweenflamestructuresobtainedwithGRI3.0(darklines),andthereducedGRIHPscheme(light

lines),foraCH4/O2counterflowdiffusionflameatP=75bar,φg=4,injectiontemperaturesTin j=200K,andstrain

rateaT=1000s−1.Normalizedmassfractionprofilesofselectedspecies(left),temperatureandheatrelease(right)are

presentedinmixturefractionspace.VerticallinesrepresentthestoichiometryZ₌0_.20.ComputedwithCantera.

mal.Thermalproperties(Table1)showthe domi-nantthermalinfluenceofthecryogenicCH4stream onthelip.Noneoftheinterfacescanbeconsidered asadiabatic.Moreover,thewallconduction char-acteristictimescaleisτw=(δlip2 ρCp/λ)=0.6 ms, whichiscomparabletothatof unsteadymotions inthefluid,justifyingtheneedforaconjugateheat transferproblem,whereboththesolidandthefluid temperaturesvaryonsimilartimescales.

3. Methane–oxygenchemistryinLREconditions FWIisexpectedtoproducelargeheatlossesat theinjectorlip, causingstrong couplingbetween heatfluxesandchemistry,andrequiringarelatively detailed kinetic scheme to account for low tem-peraturechemistrynearthelip.However,fully de-tailedmechanismsaretoocomputationally expen-siveformulti-dimensionalnumerical simulations, andveryfew reducedmechanisms havebeen de-rivedandvalidatedforCH4/O2combustionin con-ditions relevant tothose foundin LREs. Herea mechanism is derived from the detailed GRI3.0 scheme, throughareductionalgorithmproposed byPepiot-DesjardinsandPitsch[23].Theresulting scheme,denotedasGRIHP,consistsof9species,82 reactions,and7quasi-steadystatespecies.The re-ductionprocesswasalsoappliedtootherdetailed schemesadaptedforCH4 highpressure combus-tion,suchastheRAMEC[24],buttheresulting reduced mechanisms did not exhibit any signifi-cantdifferencewhencomparedondiffusion coun-terflow flames, andthe GRIHP was therefore re-tainedforthisstudy.Itwasvalidatedagainstdata fromGRI3.0 ona numberof test cases, includ-ing1D-counterflow diffusionflames overalarge rangeofpressures,strainrates,andequivalence ra-tios(Fig.2).Foralltheseflames,gaseousO2 and CH4 injection isused, because reactants are ex-pectedtobe fullyvaporizedwhen theyenterthe

flamezone[15].Theheatreleaseprofilesobtained withbothmechanisms arequitesimilar, present-ingthreereactionpeaks:aprimarypeakintherich sideof theflame(Z=0.25),asecondaryweaker peakintheleanside(Z=0.07),andan endother-miczoneintherichside(Z=0.3).Althoughthe re-ducedmechanismtendstounderestimatethe maxi-mumtemperature(−2.5%),andtooverestimatethe primaryreactionpeak(+20%),anoverall satisfac-toryagreementisobserved.Agoodagreement be-tweenGRI3.0andGRIHPisalsofoundfor auto-ignition delay times in isobaric reactors at high pressure(notshownhere).Theflamedisplayedin

Fig.2isexpectedtoberoughlyrepresentativeof combustionmodes encountered inthenumerical simulation,anditisthereforeusedinthe follow-ingof this paper asa reference forcomparison. Inparticular,theaverage volumetricheat release rate ˙0=3.5× 1011W/m3isusedtonormalizethe heatreleasecomputedinthenumericalsimulations (notedHR∗), andtheheatrelease per flame sur-facearea0=40.4MW/m2(obtainedby integra-tionovertheflamethickness)isusedtonormalize thewallheatflux(noted∗w).Thethicknessofthis flameinphysicalspaceis125μm,whichismuch largerthanthesmallestcellsize.

4. Resultsanddiscussion

Twosimulationswereperformed,thefirstone (superscriptC_{)usingconjugateheattransferatthe} injectorlip(Fig.1)thankstothecouplingstrategy describedinSection2,andthesecondone (super-scriptA_{)withadiabaticboundaryconditionsatthe} lip.

4.1. Flamerootstructure

Both simulationswere run andaveragedover 12ms,whichroughlycorrespondsto5convective

Fig.3. (1)Averagedtemperaturefieldandvelocitystreamlinesinthefluid.Averagedheatfluxmagnitudeandvector fieldinthewall.(2)Averagedheatreleaserateandvelocityvectorfieldinthefluid.Averagedtemperatureandheatflux streamlinesinthewall.(3)Scatterplotoftheflamestructureinmixturefractionspace,comparedtotheflamedisplayedin

Fig.2.SuperscriptsA_(resp.C_{)denotethesimulationwithadiabaticconditionsatthelip(resp.theconjugateheattransfer}

problem).

timesrelativetoafluidparticleintheslowdense oxygencore,orequivalentlyto20characteristic dif-fusiontimesinthesolid.Averagedfieldsofheat re-leaserate,fluidtemperature,walltemperature,and heatfluxareshowninFig.3.Theflameroot struc-tureobtainedintheadiabaticcasepresentssome distinguishablefeatures.InFig.3-(1A_)Two corotat-ingrecirculationzonesarevisible:afirstone(RA

1)is locatedinthelip’swake,whilethesecondone(RA

2) generatedbyupstreamflowseparationliesbeneath thelip,intheoxygeninjection stream.Asmaller vortexappearstobetrappedandstretchedbetween

RA 1 andR

A

2, whereapocket of hot gases (HG A₎ istrappedunderneaththelip. Thedensityisoline

ρ =0.9ρ_Oin j₂ allowstovisualizethepositionof the denseoxygencore,andthisoneappearstoretract undertheinfluenceofHGA_{,whichfacilitates} cryo-genicO2 pseudo-boiling.InFig.3-(2A) ,two dis-tinct reaction zonescanbe identified: anintense primaryreaction zone(PA

R)islocated inthelip’s wake,atadistanceof approximately1.5δlip from thelip’send,inaregionofhighshearbetweenthe twostreams.Thisstabilizationmechanismis sim-ilartothatdescribedin[6]foraLO2/GH2flame. However, unliketheLO2/GH2 flame, thepresent casealsodisplaysaweakersecondaryreactionzone (SA

R),situatedinthezoneofhighshearbetweenRA1 andRA

2.This secondaryreaction zone,anchored onto the lip within the oxygen stream, is fueled throughthepocketofhotgasesHGA_,which facili-tatescryogenicreactantsvaporization, and

there-fore theirmixing. This effectis noticeableas SA R liesclosetothestoichiometrylineZ=0.2:gaseous CH4invadesthedenseO2streamandis responsi-blefortheanchoringofthesecondaryflamefront beneaththelip.ComparedtotheLO2/GH2flame studiedin[6],thisstabilizationmechanismis prob-ablyduetoasignificantlyhighermomentumflux ratioJ=(ρFu2F)/(ρOu2O)(J=33.3here,whileJ= 0.24in[6]).Thestabilizationmechanisminthe adi-abaticcaseseemsextremelydependentonthe for-mation and the expansion of the pocket of hot gasesHGA_{.Thetemperaturesreachedbythelip} ex-tremityforthisadiabaticcaserangefrom approxi-mately300Kto3500K,andarenotrealistic.On thecontrary (see Fig. 3-(2C_{)) ,the fullycoupled} problemyieldsarelativelycold time-averagedlip temperature,which doesnot exceed450 K atits extremity. Itis also worth noting the large tem-peraturegradientacrossthelipthickness,varying from450 Kat thelowercorner of thelip(Llow) to300 K atthepoint Lup near its uppercorner. Thecoldlipinducessignificantheatlossesatthe wall(upto∗w=4.5,seeFig.3-(1C))andprevents theformationoftheregionofhotgasesHGA_seen fortheadiabaticcase,therebychangingtheentire stabilizationmechanism.Thetwomain recircula-tionzones RC

1 andR C

2 still exist,but areslightly moreseparatedthanintheadiabaticcase,sothat noarea of highshear isfound attheircommon frontier.Mixingisalsonotablyaffectedbywallheat losses:becauseoflowertemperaturesatthelip,

re-Fig.4. (1)Instantaneoustemperaturefieldandheatreleasecontoursinthefluid.Thedarklinerepresentstheρ =0.9ρin j_O2 isoline.Instantaneousheatfluxmagnitudeandvectorfieldinthewall.(2)Instantaneousheatreleaserateandvelocity vectorfieldinthefluid.ThebluelinerepresentsthestoichiometrylineZ=0.2.Instantaneoustemperaturefieldwith heatfluxstreamlinesinthewall.Subscriptsu(resp.l)denoteaninstantwhentheflamerootislocatedatthepointLup

(resp. Llow).(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionofthis

article.)

actantsvaporizationislessefficient, andgaseous CH4isnottransportedwithintheoxygenstream, asintheadiabaticcase.Onlyoneprimaryregion of reaction(PC

R)isobservedinthecoupled simu-lation.Itislocatedclosetothelipextremity,and extendsover along narrowarea spreading from thelowercornerLlowtothepointLupclosertothe uppercorner.Thissuggestsperiodicoscillationsof theflame,asdiscussedbelow.Awiderdownstream flamebrushalsotendstoindicatelargeamplitude flutteringoftheflamefrontinducedbyoscillations oftheflameroot.Detailedflamestructures(Fig.3 -(3A_)and(3C_{))inthemixturefractionspaceshow} thatwithadiabaticboundaryconditions,theflame mostlyburnsinapurediffusionmode,asits struc-tureisvery closeto thatof theone-dimensional flamepresentedinSection3.Conversely,the cou-pledproblemgivesaflamethatburnsindiffusionas wellaspartially-premixedmodes:reactionsdonot occurclosetothewallduetolowtemperatures,and reactantscanmixbeforereacting.

4.2. Flamerootdynamics

Intheadiabaticcase,theprimaryreactionzone

PA

R does not exhibit significant oscillations, and similarlytotheLO2/GH2 flameof [6],itis stabi-lizedinthehighlyshearedmixinglayerbetweenthe tworeactants streams.The weaker secondary re-actionzoneSA

R,anchoredbeneaththelipismore sensitivetoperturbationsinthecryogenicoxidizer stream, andundergoes slight intermittentcy. The flamerootobtainedwithFWIeffectsismore un-stableanddisplays strong periodicself-sustained

fluctuations.TheprimaryreactionzonePC R oscil-latesbetween twopositionsLlow locatednearthe lowerlipcorner,andLupclosertotheupperlip cor-ner.Meanwhile,theliptemperaturefieldchanges inphasewiththeflamerootmotion:atthelower cornerLlow,temperaturevariesbetween345Kand 505K,whileitvariesbetween278Kand355Kat theuppercorner.Snapshotsof temperature,heat release,wallheatflux,andwalltemperaturefields attwoinstantstupandtlow,correspondingtothetwo distinctflamerootlocations,areprovidedinFig.4.

WhenPC

R isclosetothepointLup,theflameis subjectedtoanimportantshear duetothe prox-imityof thehigh momentum CH4 stream. Asa consequence,theflamerootisarelativelystraight andunperturbedsegment,followedbylargescale vorticesdownstream.Atthesametime,wallheat lossescontributetorapidlyheatuptheupper cor-nerof thelip(∗w=8.5),resultinginarelatively homogeneoustemperaturefieldacrossthelip thick-ness(TLow=360KandTup=340K).Conversely, whenthemainreactionzonePC

R islocatednearthe lowercornerLlow,shearislessimportantduetolow velocitiesintheoxygenstream.Theflamerootis stretchedandrolledupbylargescalevortical struc-turesinthemixinglayer,anditundergoesa flap-pingmotion.Inthemeantime,thewalltemperature gradientacrossthelipthicknessreachesits max-imalvalue(TLow=490 KandTup=280K),due totheproximityof thereaction zonetothelip’s lowercorner.Inordertofurtherinvestigate cou-plingmechanismsresponsiblefortheself-sustained oscillationsaffectingsimultaneouslytheflameroot andthewalltemperature,twoquantitiesof

inter-Fig.5.Left:time-seriesoftheflamerootlocationyfr(t)andthewallhotspotlocationywh(t).Right:Fourierspectra

asso-ciatedtothesetwosignals.

estarerecordedovertime.Thefirstoneisthe ra-dial location yfr(t) of the flame root (noted fr), andthesecondoneistheradiallocationywh(t)of thewallhotspot(notedwh),defined asthepoint of maximumtemperatureinthelip(Fig.4right). Both locations are normalized by the lip thick-nessδlip.Coordinatesoriginistakenatthepoint

Lup indicated in Fig. 4.Temporal evolution and Fourier spectraof thetwo signals are shownin

Fig.5.Astrongcorrelationbetweenflamerootand wallhotspotfluctuationsappears.Bothare domi-natedbytwomodes,withafirstfundamentalmode at roughly 450 Hz and its harmonic at approx-imately 900 Hz. Thus, self-sustained oscillations affectingtheflamerootseemtoresultfroma com-plexflame/heattransfercouplingatthewall. Pres-surefluctuationswerealsorecordedatseveral lo-cationsinthedomain,andasresultingspectradid not exhibit any of the previous frequencies, the influence of acousticswas ruledout as a poten-tial cause of the unsteady flame anchoring.The wallhotspotandflamerootcoupleddynamicscan be further studied by considering the difference

y(t)=ywh(t)− yf r(t)(stillnormalizedbythelip thickness δlip). One-dimensional Dynamic Mode Decomposition[25]isusedtoisolatethemost en-ergeticcomponentsofthesignalsyfr(t)andywh(t), correspondingtothepeaksinFig.5.The result-ingphase-portraitintheplane(y(t),_y˙ (t))is dis-playedinFig.6.Theorigin(0,0)inthephase-plane represents instantswherewall hotspotandflame rootfaceeachotherandmoveatthesamespeed.A quickexaminationofthephase-portraitallowsto ruleouttheexistenceofanydistinctclosed limit-cycle,asthetrajectoryischaoticandalternates be-tweensmallorbitsaroundtheorigin(i.e.,frandwh areclosetoeachotherandhavesimilarmotion), andlargerorbits(i.e.,frandwharefarfromeach other).Mostimportantly,4cuspslocatednearthe originareclearlyvisibleonthetrajectoryshowing thatyfr≈ ywh isanunstable equilibriumposition. Afterreachingacusp,thetrajectoryissentbackto alargerorbit.Thesedynamicsareverysimilarto thatofasphereinaforceddouble-welloscillator, thecentralbumprepresentingtheunstable

equilib-Fig.6. (a)Phase-portraitofthecoupledsysteminthe plane(y(t),y˙ (t)).Redcirclesindicateunstable equi-libriumpointsatcuspsinthetrajectory.Greensquares representunstableattractingpoints.(b)Schematicview ofananalogousforceddouble-welloscillator.(For inter-pretationofthereferencestocolorinthisfigurelegend, thereaderisreferredtothewebversionofthisarticle.)

riumsituationwherewallhotspotandflameroot faceeachother.Thespherecannotreachexactlythe topofthecentralbump,andslightovershootsin itsdisplacementcauseittofallbackontheother sideandoscillateonalarger orbit.Analogously, flame rootandwall hotspot areneverexactlyin phase,andovershoots inthe flamedisplacement causethisequilibriumsituationtodivergetolarge valuesofy.Thus,thecoupleddynamicsconsists of acombinationof small-amplitudeoscillations aroundtheequilibriumstateyfr≈ ywh,and larger-amplitudeoscillationswiththeflamerootposition

yfrandthewallhotspotpositionywh decorrelated fromeachother.Physically,thisunsteadycoupling canbeexplainedbyunsteadyheatconvection phe-nomenaoccurringinthefluid.(1)Whentheflame

isstabilizednearthepointLuplargevortices down-streamof thestraightflamerootbringhot gases closetothelowercornerLlow(seeinFig.4).Thelip hotspotthenmovesfromthepointLuptoLlow,and theflamerootfollowsitsmotion.(2)Conversely, whentheflameisstabilizednearLlow,vortices con-vecthotgasestowardstheuppercorner, andthe reversemechanismtakesplace.

5. Conclusions

Two-dimensionalhigh-fidelitysimulationshave beenusedtocharacterizeflamerootstabilization mechanisms in the case of doubly transcritical LO2/LCH4 combustionproducedatthelipsof a coaxialinjector.FlameWallInteraction(FWI) ef-fectsareaccountedforbysolvingthecoupled con-jugateheattransferproblemattheinjectorlip,and byusinganappropriatedetailedkineticschemefor CH4 oxycombustion in LRE conditions. Signifi-cantlydifferentstabilizationmechanismsarefound betweensimulationsusingadiabaticwallsandthe fullycoupledcomputations.Moreprecisely,inthe former case, the flame is mainly stabilized in a sheared mixing layer between reactants streams, similarlyto LO2/GH2 flames. However,adiabatic wallsappear to facilitate cryogenic reactants va-porizationandthereforetheirmixing.Thiseffect resultsin the formation of asecondary reaction frontwithin theO2 injection stream.Conversely, thecoupledproblemexhibitsnostablemixinglayer andstrongself-sustainedoscillationsof theflame rootandof thetemperaturefieldinthelip.Both werefoundtobebimodalandtooscillateat sim-ilar frequencies, indicating a strong influence of heat transfer in the lip on the flame root loca-tion.As aresult, successiveflame root locations were observed (1) in the proximity of the CH4 stream,wheretheflameisstabilizedbyhighshear, (2) andintheproximityof theO2 stream where theflame rootisstretchedandrolled-up by vor-ticalstructures.Thechaoticcoupleddynamicsof thissystemcanbecomparedtothatof asphere slidinginaforceddouble-welloscillator.These self-sustainedoscillationsareexpectedtobeimportant fortheflameresponsetoexternalforcing,butalso fortheoverallflamestabilizationprocess.Finally, notethatthisstudyassumesperfectlysmooth injec-torwalls.Althoughthispointwasnotinvestigated, flowfeaturestakingplaceatsuchsmallscale are expectedtobehighlysensitivetowallroughness, whichcouldpotentially altersthereportedflame rootstabilizationmechanisms.

Acknowledgments

The authors thank Ariane Group, as wellas CNES,fortheirsupport.

Supplementarymaterial

Supplementarymaterialassociatedwiththis ar-ticlecanbefound,intheonlineversion,atdoi:10. 1016/j.proci.2018.05.105.

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