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Flame–wall interaction effects on the flame root
stabilization mechanisms of a doubly-transcritical
LO2/LCH4 cryogenic flame
C. Laurent, L. Esclapez, D. Maestro, G. Staffelbach, B. Cuenot, L. Selle,
Schmitt Thomas, F. Duchaine, Thierry Poinsot
To cite this version:
C. Laurent, L. Esclapez, D. Maestro, G. Staffelbach, B. Cuenot, et al..
Flame–wall
interac-tion effects on the flame root stabilizainterac-tion mechanisms of a doubly-transcritical LO2/LCH4
cryo-genic flame.
Proceedings of the Combustion Institute, Elsevier, 2019, 37 (4), pp.5147-5154.
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To link to this article: DOI:10.1016/j.proci.2018.05.105
URL:
https://doi.org/10.1016/j.proci.2018.05.105
This is an author-deposited version published in:
http://oatao.univ-toulouse.fr/
Eprints ID: 22926
To cite this version: Laurent, Christophe
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
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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
a,∗,
L.
Esclapez
a,
D.
Maestro
a,
G.
Staffelbach
a,
B.
Cuenot
a,
L. Selle
b,
T.
Schmitt
c,
F.
Duchaine
a,
T.
Poinsot
ba CERFACS,42AvenueGaspardCoriolis,ToulouseCedex131057,France b IMFT,AlléeduProfesseurCamilleSoula,Toulouse31400,France
cLaboratoireEM2C,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 /O2 cryogeniccombustion,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 /GH2 shear-coaxialflames wascharacterizedexperimentally[9–11],andthese resultswerelaterrecoverednumericallywithLarge EddySimulation(LES)[12].LO2 /CH4 flames un-der various injection conditionswere also exam-inedexperimentally[13]andnumerically[14].
Mostpreviousresearcheffortsfocusedon super-criticalflameswhereoxygenwasinjectedinadense transcriticalstate,andfuel(eitherH2 orCH4 )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 )/(ρOu2 O)=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
isother-Fig.2.ComparisonbetweenflamestructuresobtainedwithGRI3.0(darklines),andthereducedGRIHP scheme(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-tweenGRI3.0andGRIHP isalsofoundfor 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× 1011 W/m3 isusedtonormalizethe 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 andRA2 , whereapocket of hot gases (HGA) istrappedunderneaththelip. Thedensityisoline ρ =0.9ρO in 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 /GH2 flame. 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 CH4 invadesthedenseO2 streamandis responsi-blefortheanchoringofthesecondaryflamefront beneaththelip.ComparedtotheLO2 /GH2 flame studiedin[6],thisstabilizationmechanismis prob-ablyduetoasignificantlyhighermomentumflux ratioJ=(ρFu2 F)/(ρOu2 O)(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
theformationoftheregionofhotgasesHGAseen
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 jO2 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.
References
[1]V.Yang,Proc.Combust.Inst.28(1)(2000)925–942.
[2]J.Bellan,Prog.EnergyCombust.Sci.26(4)(2000) 329–366.
[3]B.Chehroudi,D.Talley,E.Coy,Phys.Fluids14(2) (2002)850–861.
[4]W. Mayer, J. Telaar, R. Branam, G. Schneider, J. Hussong, Heat Mass Transf. 39 (8–9) (2003) 709–719.
[5]J.C.Oefelein,V.Yang,J.Propuls.Power14(5)(1998) 843–857.
[6]J.C. Oefelein, Proc. Combust. Inst. 30 (2) (2005) 2929–2937.
[7]J.C.Oefelein,Combust.Sci.Technol.178(1–3)(2006) 229–252.
[8] L.Vingert,M.Habiballah,P.Vuillermoz, S. Zur-bach,51stInternationalAstronauticalCongress,Rio deJaneiro,Brazil(2000).
[9]S.Candel,P.Herding,P.Scouflaire,etal.,J.Propuls. Power14(5)(1998)826–834.
[10]M. Juniper, A.Tripathi,P.Scouflaire,J.C.Rolon, S. Candel, Proc. Combust. Inst. 28 (1) (2000) 1103–1109.
[11]M.Juniper,S.Candel,J.Propuls.Power19(3)(2003) 332–341.
[12]T.Schmitt,L.Selle,B.Cuenot,T.Poinsot,C.R.Méc.
337(6–7)(2009)528–538.
[13]G.Singla,P.Scouflaire,C.Rolon,S.Candel,Proc. Combust.Inst.30II(2)(2005)2921–2928.
[14]T.Schmitt,Y. Méry,M.Boileau,S.Candel,Proc. Combust.Inst.33(1)(2011)1383–1390.
[15]G. Lacaze, J.C.Oefelein,Combust.Flame 159(6) (2012)2087–2103.
[16]V.Moureau,G.Lartigue,Y.Sommerer,C. Angel-berger,O.Colin,T.Poinsot,J.Comput.Phys.202(2) (2005)710–736.
[17]G.Soave,Chem.Eng.Sci.27(6)(1972)1197–1203.
[18]T.Schmitt,L.Selle,A.Ruiz,B.Cuenot,AIAAJ.48 (9) (2010)2133–2144.
[19]F.Duchaine,S. Mendez,F.Nicoud,A. Corpron, V. Moureau,T.Poinsot,C.R.Méc.337(6–7)(2009) 550–561.
[20]R. Mari, B. Cuenot, J.P. Rocchi, L. Selle, F. Duchaine,Combust.Flame168(2016)409–419.
[21]F.Duchaine,S.Jauré,D.Poitou,etal.,Comput.Sci. Discov.8(1)(2015)015003.
[22]T.J.Poinsot,S.Lele,J.Comput.Phys.101(1)(1992) 104–129.
[23]P.Pepiot-Desjardins,H.Pitsch,Combust.Flame154 (1) (2008)67–81.
[24]E.L. Petersen, D.M. Kalitan, S. Simmons, G. Bourque, H.J. Curran, J.M. Simmie, Proc. Combust.Inst.31(1)(2007)447–454.