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

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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

Section2,togetherwiththedescriptionoftheLES dataset. The limit cycle predicted in the LES is analyzedinSection3.Section4isdevotedtothe description of the global structure of the flame

response and the relative contributions of the

two dominant modes are discussed. Finally in

Section5thephysicalmechanisms 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.

ThecomputationaldomainshowninFig.1is

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 inFig.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 discussedinSection5.

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

4. Mapsofflameresponse

Theobjectiveofthissectionistoquantifythe unsteadyresponseoftheflamesanddeducemaps ofthecontributionofthetwoeigenmodes identi-fiedinSection3.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

inFig.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_{.UsingEq.(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 inFig.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

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.

Figure9suggeststhepresent 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(redregionsinFig.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-ceivedfundingfromtheEuropeanResearch

Coun-cil under the European Union’s Seventh

Frame-work Programme (FP/2007-2013) / ERC Grant

AgreementERC-AdG319067-INTECOCIS.

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