Study of flame response to transverse acoustic modes from the LES of a 42-injector rocket engine

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Study of flame response to transverse acoustic modes

from the LES of a 42-injector rocket engine

Annafederica Urbano, Quentin Douasbin, Laurent Selle, Gabriel Staffelbach,

Bénédicte Cuenot, Thomas Schmitt, Sébastien Ducruix, Sébastien Candel

To cite this version:

Annafederica Urbano, Quentin Douasbin, Laurent Selle, Gabriel Staffelbach, Bénédicte Cuenot, et

al.. Study of flame response to transverse acoustic modes from the LES of a 42-injector rocket

engine. Proceedings of the Combustion Institute, Elsevier, 2017, vol. 36 (n° 2), pp. 2633-2639.

�10.1016/j.proci.2016.06.042�. �hal-01551126�

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To cite this version : Urbano, Annafederica and Douasbin, Quentin

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

Schmitt, Thomas and Ducruix, Sébastien and Candel, Sébastien Study

of flame response to transverse acoustic modes from the LES of a

42-injector rocket engine. (2017) Proceedings of the Combustion

Institute, vol. 36 (n° 2). pp. 2633-2639. ISSN 1540-7489

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Study

of

flame

response

to

transverse

acoustic

modes

from

the

LES

of

a

42-injector

rocket

engine

A.

Urbano

,

Q.

Douasbin

,

L.

Selle

,

G.

Staffelbach

,

B.

Cuenot

,

T.

Schmitt

,

S.

Ducruix

,

S.

Candel

a_{SafranTech,E&P,RuedesJeunesBois,Châteaufort,CS80112,78772Magny-Les-Hameaux,France} b_{Université deToulouse,INPT,UPS,IMFT(InstitutdeMécaniquedesFluidesdeToulouse),AlléeCamilleSoula,}

Toulouse31400,France

c_{CNRS,IMFT,Toulouse,31400,France}

d_{CERFACS,42AvenueGaspardCoriolis,ToulouseCedex0131057,France}

e_{LaboratoireEM2C,CNRS,CentraleSupélec,Université Paris-Saclay,GrandeVoiedesVignes,Chatenay-Malabrycedex}

92295,France

Abstract

TheLarge-EddySimulationof areduced-scalerocketengineoperatedbyDLRhasbeenconducted.This configurationfeatures42coaxialinjectorsfedwithliquidoxygenandgaseoushydrogen.Foragivensetof injectionconditionsthecombustorexhibitsstrongtransversethermo-acousticoscillationsthatareretrieved bythenumericalsimulation.ThespatialstructureofthetwomainmodesobservedintheLESisinvestigated through3DFourieranalysisduringthelimitcycle.Theyarerespectivelyassociatedwiththefirsttransverse andfirstradialresonantacousticmodesofthecombustionchamber.Thecontributionsofeachindividual flametotheunsteadyheatreleaserateandtheRayleighindexarereconstructedforeachmode.These contri-butionsareinbothcaseslowinthevicinityofvelocityanti-nodesandhighnearpressureanti-nodes. More-overitisnoticedthatthesepressurefluctuationsleadtolargevelocityoscillationsinthehydrogenstream. Fromtheseobservations,adrivingmechanismfortheflameresponseisproposedandvaluesforthegainand phaseoftheassociatedflametransferfunctionareevaluatedfromtheLES.

Keywords: Large-EddySimulation;Rocketpropulsion;Transversecombustioninstability;Flametransferfunction; Rayleighindex

∗_{Corresponding author at: Institut de Mécanique}

des Fluides de Toulouse, Allée Camille Soula, 31400 Toulouse,France.Fax:+33534322994.

E-mailaddress:laurent.selle@imft.fr(L.Selle).

1. Introduction

The occurrence of combustion instabilities has plagued many development programs for high-performance propulsion systems [1–4] . In theaerospace industry, oneof the moststriking examplesmaybethedevelopmentoftheF1engine

fortheApollomission,whichrequired1332 full-scalehot-firetestsand108injectordesignchanges before meeting both stability and performance requirements [5] . The cost associated with such trial-and-errorprocedurecanbeprohibitive justify-ingthesearchformethodsallowingtheprediction ofstabilitycharacteristicsatthedesignstage.

Withgrowingcomputationalresourcesavailable toresearchersandengineers,andthedevelopment of HighPerformance Computing,it istimely to address this problem with numerical tools that simultaneously solve for turbulence, acoustics and combustion. Indeed, high fidelity modeling strategies such as Large-Eddy Simulation (LES) havehadconsiderablesuccessesinpredicting un-stableoperatingpointsforgasturbinecombustors

[6,7] orgaseouscoaxialinjectors [8–10] .Withthe recentdevelopmentofLESfortranscriticalflows

[11–14] ,high-pressureliquidrocketenginestability cannowbenumericallystudied [15] .Nevertheless, these methods are usually too cumbersome to allowexplorationsofthewholerangeofoperating conditions.Moreover,itmightevenbeawasteof resourcesto systematicallyuseLES because itis likelythattherearegenericfeaturespertainingto theinjectionunits andsystemortothe combus-tionchamber,whichdonotrequireahigh-fidelity simulationtobepredicted.

Onealternativetothis“bruteforce” approach isthejointuseof aflame-responsemodelandan acousticsolver [16] .This has beendemonstrated in a generic configuration [17] but also in more complex geometries including turbulent flames

[18] .Thisstrategycanhelptheanalysisofunstable modes and deliver good predictions of stability maps[19] .Thishasalsobeentriedforrocketengine configurations [9,20] withsomesuccess.However, twokeyingredientsarerequiredasinputsforthis approach:

1. A baseline flow field. When solving the Helmholtz equation, the field of speed of sound is needed. If linearizedEuler equa-tions arechosen,themean velocity fieldis alsorequired.

2. Amodelfortheresponseoftheflameto acous-ticperturbations.Thedrivingmechanismfor the amplification of acoustic perturbations involves the coupling with unsteady heat release rate fluctuations. This often arises through a complex mechanism combining nonlinearfluidmechanicsandtheflame re-sponsetoincidentacousticperturbation.A globalmodelforthiscomplexinteractionis thenrequiredtofeedtheacousticsolver. Thebaselineflowcanusuallybeobtainedfrom lightweightcomputations ortheory,but in some caseshigh-fidelitysimulationsarerequiredbecause it directly influences the eigenfrequency values. However,theflameresponseisvirtuallyimpossible toderivefromtheoreticalconsiderations,exceptin

somesimplecases,anditsaccuratedetermination iscentraltothepredictionof stabilitymaps. Ex-perimentaldeterminationsareeasilyobtainedfor atmosphericpressure systemsbutthere are tech-nicaldifficultieswhen operatingathigh-pressure. This iswhere thehigh-fidelity numerical simula-tionsofthe“bruteforce” approachcanbeofhelp. Inwhatfollows,calculations willnot be usedto derivestability maps butformore modest goals: (1)Understandingphysicalmechanismsthatdrive theflameresponsesand(2)Obtainaquantitative evaluationofthisflameresponse.

Theobjective of thepresent work istousea time-resolveddatasetof3Dsolutionsobtainedby LESforthestudyofinjectorresponseduringthe limitcycleofacombustioninstability.High-order numericsandstateoftheartsubgrid-scalemodels ensurethehighfidelitycharacterofthedatabase. Also,withtheintenttomodeltheunsteadyflame response to be fed into an acoustic solver, the questionofwhichflowvariablesaremostrelevant isaddressed.ItisindicatedbyYangandAnderson

[5, chap. 1, p. 9] that thephysical and chemical processesintheimmediatevicinityofthechamber backplanearegenerallyquitesensitivetothe trans-versevelocityperturbationsparalleltothatplane andlesssusceptibletotheunsteadymotionsacting inthemainflowdirectionatright anglestothat plane.Ontheotherhand,muchoftheworkinthis domain(e.g.Croccoetal., [21] )emphasizeseffects of unsteady pressure as the input for the flame responseandthishasledtosomemeaningful re-sults.Itisthusinterestingtoidentifytheprocesses thatfeedenergyintothecouplingmodesanddrive the unstable oscillations and more specifically compareeffectsoftransversevelocitieswiththose of pressureperturbations inthe near vicinity of theinjectorbackplane.Thiswillbedonehereby makinguseofthehigh-fidelitysimulationdataset. The configuration is briefly presented in

Section 2 ,togetherwiththedescriptionoftheLES dataset. The limit cycle predicted in the LES is analyzedin Section 3 . Section 4 isdevotedtothe description of the global structure of the flame response and the relative contributions of the two dominant modes are discussed. Finally in

Section 5 thephysicalmechanisms thatdrivethe unsteadyflameresponseareidentified,withafocus onthespecificinjectorsthatplaythemost impor-tant role in the destabilizationof thetransverse mode.

2. ConfigurationandLESdataset

The configuration is a reduced-scale rocket engine called BKD operated at DLR Lampold-shausen consisting of a cylindrical combustion chamber, 8 cm in diameter, fed by 42 coaxial injectors and closed by a choked nozzle. The injectionplatepatterncomprisesthreeconcentric rings of respectively 6, 12 and 24 injectors. The

propellantsareoxygenandhydrogenandtheBKD is typically operated in the transcritical regime where liquid oxygen at subcritical temperature entersthechamberthatisatsupercriticalpressure. A description of the apparatuscan be foundin theworkof Gröningetal. [22–24] .Theoperating conditions correspond toan unstable load point referredtoasLP4forwhichthepressurechamber is pc=80 bar andthe thermal power assuming

completecombustionisaround86MW.The adi-abatic burnt-gas temperature assuming complete combustionisTc=3627K.

The LES has been carriedout with the real-gas flow solver AVBP-RG [25] jointly developed by CERFACS, IFPEN and EM2C. A two-step Taylor–Galerkin scheme called TTG4A, is used, whichisthirdorderinspaceandfourthorderin time [26,27] . The solver accounts for multicom-ponent real-gas thermodynamics and transport

[28,29] .TheWallAdaptingLinearEddy(WALE) model is used to close the subgrid stress tensor

[30] andan eddy-diffusivityapproach is adopted for thermal and species subgrid contributions (constantturbulentPrandtlandSchmidtnumber: Prt=0.6 ,Sct=0.6). A infinitely-fast chemistry

model [13] , relying on the assumption of local chemical equilibriumandaβ-pdf descriptionof thefiltered mixturefraction Z, isadopted. Four speciesareconsidered(H2,O2,OHandH2O)and

sourcetermsarecomputedfollowingthemethod describedin [13] .Thismodelimpliesthatthe an-choringpointoftheflameisatthelocationwhere O2andH2streamsmeetandcanthereforenot

oscil-late.Consequently,theinfluenceofflame-root mo-tionontheflameresponseisnottakenintoaccount inthissimulation.Specificmassflowratesand tem-peratureofO2andH2(m˙O2=5.75kg.s−1,m˙H2=

0.96 kg.s−1, TOin j2 =111 K and T

in j

H2 =96 K) are

imposedatthedomesmanifoldsinletsusing char-acteristic treatment of the boundary conditions

[31] ,adaptedtoreal-gasthermodynamics.The out-letnozzleischoked,requiringnoboundary treat-ment.Thewallsareassumedtobeadiabaticandare treatedasno-slipboundariesintheinjectorsandas slip-boundariesinthechamberandinthedomes.

Thecomputationaldomainshownin Fig. 1 is discretizedwitha70Melementmesh.Thetypical meshresolutioninthezonewheretheflamesare established is  =50 μm. The resulting CPU requirementsare100,000honaBlueGene Qfor the simulation of 1 ms (which corresponds to abouttentimestheperiodofthetypicaloscillation at the first transverse mode). A typical run is performedinparallelon16,384cores sothatthe restitutiontimeisreasonabledespitethesignificant computationalburden.1

1 _{Because theAVBP solver can make use of}

hyper-threadingonBlueGeneQarchitectures,thereare4MPI

Fig.1. Overviewof thecomputationaldomainforthe BKD(top). Transverse (bottomleft) and longitudinal (bottomright)cutsofinstantaneoustemperaturefield.

PSD

[dB/Hz]

f [kHz]

Fig.2.Powerspectraldensityofpressurefluctuationsat thechamberwall5.5mmdownstreamtheinjectionplate.

AlimitcycleisreachedintheLESandthe dy-namicsofthesystemiscomputedoveraperiodof 7.5ms.Adetailedpresentationof thissimulation isgivenin [15] andtheaimofthepresentworkis toperformanin-depthanalysisof thelimitcycle andanalyzetheflameresponse.Forthispurpose, 200snapshotsof thefull3Dsolutionweresaved over2msofthelimitcycle(between5and7ms), which corresponds to 330 Gb of data. Acoustic andcombustionfluctuationsareanalyzed,making useofFouriertransformofthe3Dfields(3D-FT) atthefrequenciesofinterest.

3. Descriptionofthelimitcycle

During the limit-cycle predicted by the LES, pressurefluctuationsof verylargemagnitudeare recorded. The rms value reaches prms=0.15pc

which corresponds to 10.7 bar. Pressure spikes reaching+44bararesometimesobserved.

Thepower spectraldensity(PSD)of pressure fluctuationsatasensorplacedonthechamberwall 5.5mmdownstreamtheinjectorplateisdisplayed in Fig. 2 .Therearetwo dominantfrequenciesat f1=10,700 Hzand f2=21,400 Hz,which are

processespercoreresultinginatotalof65,536MPI pro-cessesforthiscomputation.

Fig.3. Spatialstructureofthepressurefluctuationsfor thetwodominantfrequenciesofFig.2fromthe3D-FT of200instantaneousLESfields.

close andwithin 5%of experimentally observed frequencies [22] .

Itispossibletoextractthepressuredistributions correspondingtothesetwofrequenciesbytaking the Fouriertransform of the200 pressure fields accumulatedinthedataset.Whilef1 corresponds

to the first transverse mode (labeled 1T) of the chamber(Fig. 3 ),themodeshapeatf2resembles

thefirstradialmode(labelled 1R).Inbothcases these chamber modes are strongly coupled with the oxygen injectors where longitudinal fluctua-tionsareobserved.Thehydrogeninjectorsdonot seemtobeaffectedbythepressurefluctuationsin thechamber,whichisconsistentwiththeirsmall radii(of theorderof 0.25mm).Nevertheless,an examinationof the velocityfields fromthe PSD (not shown) indicates thatthe hydrogen stream, at the injectorexhaust andfurther downstream, experiencesstrongvelocityfluctuationsbecauseof theeigenmodesinthechamber.Thismechanismis discussedinSection 5 .

Finally,themodestructuresof Fig. 3 suggest thatinjectorslocatedonthenodallineof the1T mode will mostly experience transverse velocity fluctuations.Similarly,the1Rmodewillproducea transverseacousticvelocityonthesecondinjector ring.

4. Mapsofflameresponse

Theobjectiveofthissectionistoquantifythe unsteadyresponseoftheflamesanddeducemaps ofthecontributionofthetwoeigenmodes identi-fiedinSection 3 .Tothispurpose,itisconvenientto defineboxesthatisolateindividualflames.Firstthe threeringsareseparatedbycylindricalboundaries, thenneighboringflamesbyradialplanes.Allthese boundaries are chosen to be at equal distances fromneighboringinjectors.Variousquantitiescan then beintegrated in theseboxesoverthewhole lengthofthecombustionchamber.

Fig. 4. Mapsof unsteady heat release rate integrated aroundeachflame.

This processingmethod isapplied to the un-steady heat release rate, q, extracted from the 3D-FTof bothmodes.Theresulting maps of q for each injector and both modes are displayed in Fig. 4 .Regardingthe1Tmode,theflamesthat exhibitthegreatestresponse arethose locatedin the region where the pressure fluctuations reach theirmaximum(cf.Fig. 3 ).Onthenodallineofthe pressurefield,theflameresponseisfoundtonearly vanish. This indicates that the flames respond weaklyto the transversevelocity fluctuations of the1Tmode.Similarconclusionsaredrawnfrom themapofqinthe1Rmode:theinnerandouter ring,correspondingtopressureantinodes,respond stronglyandoutofphase,whilethemiddleinjector ringisvirtuallyinactive.

Inordertoquantifytheimpactofthese fluctu-ationsonthegrowthof theinstability,itisuseful toconsidertheRayleighindex,definedas:2

R= 1 T γ − 1 γ p0  T  V p(t)q(t)dVdt (1) whereT is atime span thatcovers at least one periodoftheoscillationsandVavolumethat con-tainsalltheflames.ThistotalRayleighindex ac-countingforallpressureandheatreleaserate per-turbationisR=125kWintheLES.Thispositive valueisconsistentwiththefactthatcombustionis drivingtheinstabilityandalimitcycleisreachedin theLES.Withtheintenttoseparatetheimpactof thetwodominanteigenmodes,theRayleighindex, Ri,ofeachindividualmodecanbeevaluatedas:

Ri = γ − 1 2γ p0  V |q˜i||p˜i|cos(φq˜i− φp˜i)dV (2)

where p˜i (respectively q˜i) is the 3D-FT of

pres-sure (respectively heat release rate) fluctuations. Thephasesφ correspondtothedefinitionwhere

˜

p=|p˜|eiφ_{.Using Eq. (2) ,therespective}

contribu-tionsofthe1Tand1RmodesareR1=42.2kW

andR2=8.8kW.Itfollowsthatbothmodesare 2 _{This definition would not be consistent with the}

acousticenergyconservationinarealgasbecausethe nor-malization(_{γ − 1)/}(_{γ p}0)correspondstoaperfectgas.

Here,weusedconstantvaluescorrespondingtotheburnt gasesinthechamber.Theevaluationoftheresulting dis-crepanciesislefttofurtherstudies.

Fig.5.MapsofindividualflamesRayleighindexforboth 1Tand1Rmodes,normalizedbythetotalRayleighindex ofthechamber.

drivingthe instability andthat the1T mode ac-countsfor33.8%ofthedestabilizationwhilethe1R modecontributionamountsto7.0%ofthetotal.

Onemaynowfocusonthecontributionof in-dividualinjectorsbyexaminingmapsofRayleigh index integrated around each injector. Figure 5

presentsthecontributionsofthe1Tand1Rmodes, normalizedbythetotalRayleighindex.

First,regardingthe1Tmode,theshape of q maps isrecovered(cf. Fig. 4 ) with injectorsat a pressureantinodecontributingthemostandthose on the nodal line being virtually inactive. The maximum contributionof an individual injector is1.8%.Forthe1Rmode,onlythesixinjectors of theinnerringhaveasignificantcontribution, withamaximumof0.6%.Itisinterestingtonote thatdespitethehighlevelsof qontheouterring (cf. Fig. 4 ),theirphasedoesnotseemtoallowa significantcontributiontotheinstability.

5. Individualinjectordynamics

Thequestionthatonemaynowaddressisthat of thephysicalmechanisms drivingtheunsteady responseof thesecoaxialflames.Thefocusisset onthe1Tmode,whichcontributesthemosttothe drivingprocess,andtwotypicalflamesaresingled out:

• AnA-flamelocatedatapressureantinode. It was shown in Figs. 4 and 5 that these flamesrespondstronglytothebulkpressure fluctuationinthechamber.

• AnN-flamelocatedatapressurenode.These flames experience little pressure variations butastrongtransversevelocityfluctuation. It was shown in Figs. 4 and 5 that they respondweaklyintermsofheatreleaserate fluctuation.

The heatrelease ratefluctuations q,averaged overavolumecomprisingeachflamearecompared in Fig. 6 ,whereq0isthetimeaveragedheatrelease

of the flame. As expected, the response of the A-flame is larger, consistently with Fig. 4 . This confirms that these coaxial diffusion flames are moresensitivetopressurefluctuationsthantothe transversevelocityinducedbytheeigenmode.

q’/q

0

t [ms]

A N

Fig.6.Comparisonofheatreleaseratefluctuationsfora:

A-flame; N-flame. p’ recess u’H2 u’O2 u’_O2 u’H2

Fig.7. Schematicofthecoaxialinjectorandreference surfacesfortheextractionofvelocitiesandpressure fluc-tuationsusedtoevaluatetheflameresponses.

u’/u

m

t [ms]

Fig.8. Velocityfluctuationatthelocationoftherecess:

u_H

2; u

 O2.

Onemaynowproceedwithadetailedanalysis oftheA-flame.Aschematicrepresentationofthe recessedcoaxialinjectorof theBKDisshownin

Fig. 7 . Whenan acousticmodeis excited inthe combustion chamber,the injectorof an A-flame experiencesa back-pressurefluctuation,p,at its exitplane, which in turngenerates velocity fluc-tuations inbothpropellantstreams.The velocity fluctuations,uH₂ anduO₂,averagedovertheir

re-spectivecrosssection(anannulusforH2andadisk

forO2),atthelocationoftherecessarecompared

inFig. 8 .Theyarenormalizedbythemeanvelocity um=(u0,H2+u0,O2)/2wheresubscript0indicatesa

timeaveraging.Thischoiceforthenormalizationis supportedbythefactthattheflowdownstreamthe coaxialinjectorresemblesapulsatedmixinglayer. It shows which stream oscillates the most with respect to the mean velocity. When each stream is normalized by its own mean velocity, which is muchlower forthe dense oxygen,the relative

fluctuationlevelsarecomparableandaround10%. It appears that the O2 velocity fluctuations are

negligiblecomparedtothoseofH2.Bothstreams

experiencethesamepressureperturbationbutthe correspondingudependsontheimpedancewhich isrelatedbothtogeometric(arearatios)and ther-modynamic (compressibility) effects. Specifically, the u amplitude is inversely proportional tothe characteristicimpedance of thegas,whichisthe product of the densityand speed of sound: ρc. The thermodynamic conditions at the location of the recess are: (ρc)O2=710

5 _kg.m−2_.s−1 _and

(ρc)H2=1.8 10

4 _kg.m−2_.s−1_{, which is 40 times}

higherforO2 thanforH2.Thispossiblyexplains

why the velocity fluctuations in the H2 stream

dominateinthepresentconditions.

Toquantify thecorrelation between velocities orpressureandheatreleaseperturbationsonemay calculatethenormalizedcrosscorrelationdefined by:

rf g=

( f g)(τ ) σfσg

(3) whereσ isthestandarddeviation.Themaximum correlationbetweenpandqisrpq=80%whileit

is67%betweenu_H

2andq

 _{andfallsdownto47%}

between uO2 andq

_{.These observations are}

con-firmed by singleinjectorsimulations (notshown here),wheretheflameresponsesinducedbyH2or

O2streamsfluctuationsarecompared.Forcingthe

individualpropellantsvelocitiesatthelevel mea-suredinthefull engine,i.e.around 10%of their mean,yieldsamuchweakerflameresponseforO2

than for H2. Nevertheless bulkpressure

fluctua-tionsattheoutletoftheinjectoralsotriggered sig-nificantlevelsofq.

From these observations we can assume that uO2isnotthemostrelevantinputvariableforthe

flame response. One may then speculate that q isdrivenbyuH₂ throughtheforcingof theshear

layer, generating unsteady coherent structures affecting both wrinkling and local stretch of the flame eventually leading to heat release rate fluctuations.Amechanismfortheflameresponse is now proposed, based on the above rationale. Hydrogenvelocityfluctuationsaresupposedtobe centralinthismechanismbutthevalidationofthis hypothesis requires additional tests. Mechanisms involvingadirectresponsetopressurefluctuations, forexample,shouldalsobeconsidered.The mech-anismissummarizedinFig. 9 ,wherethetemporal evolutionofp,uH2andq

_{,extractedfromtheFT}

atthe1Tfrequency,areshownovertwocyclesof theinstability.Threetimedelaysareidentified:

• ThedelayofuH₂withrespecttop:τup.

• Thedelayofqwithrespecttou_H

2:τqu.

• Thedelayofqwithrespecttop:τqp.

Figure 9 suggeststhepresent scenarioforthe A-flameresponse: thepressure fluctuationatthe

u’H2 [m.s -1] p’ [bar] q’ [kW] up qu qp

Fig.9. Fluctuationsovertwoperiodsof the1Tmode (T=1/ f1)foranA-flame: p; uH2; q

_.

Ref-erencesurfacesareindicatedinFig.7.

injectoroutletgeneratesahydrogenvelocity fluctu-ationafteratimeτup,whichdrivestheshearlayer

and subsequently heat release rate fluctuations withadelayτqu.Theoveralldelayτqp=τup+τqu

issuchthatpandqarealmostperfectlyinphase, resultinginapositiveRayleighindex.Whileτupis

mainlyacousticbynature,τqurepresentsthetime

forhydrodynamicsandcombustiontorespondto theunsteadyshear.

Finally,theoverallresponseofanA-flamecan bequantifiedbythegain,nandtimedelay,τ ofq versusp:

n= |q˜|/q0 |p˜|/p0

τ = φq˜− φ˜p

2π f (4) The present datasetisused to computenand τ forthe A-flames of the 1Tmode. Here we give averagedvaluesforthe8outerA-flamesthathave thehighestRayleighindex(redregionsin Fig. 5 ): n=1.1andτ =0.9T.

6. Conclusions

Inthisarticle,theLarge-EddySimulationofa 42-injectorreduced-scalerocketengineisusedto analyzethelimitcycleofacombustioninstability. The post-processing of a time-resolved dataset of3Dsolutionsallowstoisolateindividualflame dynamicsaswellastheinfluenceofdifferent eigen-modesof thechamber.Inthisconfigurationtwo chambermodes dominate,onewith atransverse shape andthe other with aradial structure.For bothmodes,themagnitudeof theflameresponse ismaximumatpressureantinodeswhiletheflames locatedatapressurenoderespondweakly, suggest-ingthat the lateral motion caused bytransverse velocityfluctuations doesnot effectivelyfeed en-ergyintoacousticsforsustainingthisinstability.A mechanismisproposedinwhichthebulkpressure variationattheinjectoroutletgeneratesunsteady shear through thevariation of the hydrogen ve-locity, ultimately resulting in heat release rate fluctuations.Formodelingpurposes,itissuggested toconsiderthefluctuatingpressureintheinjection planeastherelevantinputfortheflameresponse.

This option has been considered since the early studies on transverse combustion instabilities in rocketengines [21] anditreceiveshereadditional supportfrom3Dunsteadynumericalsimulation.

Acknowledgments

Thisinvestigationwascarriedoutinthe frame-work of the French–German REST program initiated by CNES, DLR, Astrium andSnecma. Support providedby Safran (Snecma) theprime contractoroftheArianerocketpropulsionsystem isgratefullyacknowledged.

Allgeometrical,operational,andmeasurement datarelatedtotheBKDwas kindlyprovided by DLR Lampoldshausen. The authors are partic-ularly gratefulto Stefan Gröning andcolleagues who performed the experiments and formulated thetestcase.

The authorsacknowledgePRACEfor award-ing usaccessto resourceFERMI basedin Italy at Cineca. This work was granted accessto the high-performancecomputingresourcesof IDRIS undertheallocationx20152b7036madebyGrand EquipementNationaldeCalculIntensif.

The research leading to these results has re-ceivedfundingfromtheEuropean Research Coun- cil under the European Union’s Seventh Frame- work Programme (FP/2007-2013 ) / ERC Grant AgreementERC-AdG 319067-INTECOCIS .

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