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Nonlinear thermoacoustic mode synchronization in
annular combustors
Jonas P. Moeck, Daniel Durox, Thierry Schuller, Sébastien Candel
To cite this version:
Jonas P. Moeck, Daniel Durox, Thierry Schuller, Sébastien Candel. Nonlinear thermoacoustic mode
synchronization in annular combustors. Proceedings of the Combustion Institute, Elsevier, 2018, 37
(4), pp.5343-5350. �10.1016/j.proci.2018.05.107�. �hal-02135736�
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Moeck, Jonas P. and Durox, Daniel and Schuller, Thierry and Candel, Sébastien Nonlinear
thermoacoustic mode synchronization in annular combustors. (2018) Proceedings of the
Combustion Institute, 37 (4). 5343-5350. ISSN 1540-7489
OATAO
Nonlinear
thermoacoustic
mode
synchronization
in
annular
combustors
Jonas
P.
Moeck
a,b,∗,
Daniel
Durox
c,
Thierry
Schuller
c,d,
Sébastien
Candel
caDepartmentofEnergyandProcessEngineering,Norwegian UniversityofScienceandTechnology,Trondheim7491, Norway
bInstitutfürStrömungsmechanikundTechnischeAkustik,TechnischeUniversitätBerlin,Berlin10623,Germany cLaboratoireEM2C,CNRS,CentraleSupélec,Université Paris-Saclay,Gif-sur-Yvettecedex92292,France dInstitutMécaniquedesFluidesdeToulouse,Université deToulouse,CNRS,INPT,UPS,Toulouse31400,France
Abstract
Nonlinearcouplingbetweenazimuthalandaxisymmetricmodesinannularcombustorsisstudied analyti-cally.Basedonthethermoacousticwaveequation,amodelfeaturingthreenonlinearlycoupledoscillatorsis derived.Twooscillatorsrepresentthedynamicsofanazimuthalmode,andthethirdaccountsforthe axisym-metricmode.Aslow-timesystemfortheevolutionofthemodeamplitudesandphasesisobtainedthrough theapplicationofthemethodofaveragin g.Theaveragedsystemisshowntoaccuratelyreproducethe solu-tionsofthefulloscillatormodel.Analysisofthisfive-dimensionaldynamicalsystemshowsthatastanding azimuthalmodemaysynchronizewithanaxisymmetricmode,providedthattheirindividualresonance fre-quenciesandgrowthratesaresimilar.Thisphase-coupledtwo-modeoscillationcorrespondstotheso-called slantedmode,observedinrecentexperimentsinvolvinganannularmodelcombustionchamber.Quantitative conditionsfortheoccurrenceofmodesynchronizationarederivedintermsofthegrowthrateratioanda frequencydetuningparameter.Theanalysisresultsarefoundtobeconsistentwithexperimentalobservations oftheslantedmode.
Keywords: Combustioninstability;Azimuthalmode;Nonlinearsynchronization;Annularchamber;Slantedmode
∗Correspondingauthorat:DepartmentofEnergyand
ProcessEngineering,NorwegianUniversityof Science andTechnology,Trondheim7491,Norway.
E-mail addresses: jonas.moeck@ntnu.no,
jonas.moeck@tu-berlin.de(J.P.Moeck).
1. Introduction
Thermoacoustic instabilities occur in many technicalapplicationswhereheatisaddedthrough combustion,prominentexamplesbeingstationary gasturbinesforpowergenerationandaero-engines
[1,2].Insimplified laboratoryconfigurationsthat hostasingleflame,acousticwavespropagateonly
in the axial direction and the associated modes arereadilyobtainedfromelementarycalculations. Combustion chambers in the applications men-tionedabove,however,featureanannular geome-try,inwhichmultipleflames,typicallymorethan 10,aredistributedcircumferentially.Theseannular combustionchambershost,inaddition,azimuthal modes,forwhichthedominantpressurevariation occursalongtheangularcoordinate.Infull-scale engines,combustioninstabilitiesarethusoften ob-served to couplewith the lower-order azimuthal modes[3,4].
Overthelastdecade,azimuthalinstabilitiesin annularchambershavebeenintenselystudiedby experimental andnumerical means [5–8] andby using analytical methods [9]. Many elementary propertiesofazimuthalinstabilitymodesarewell understoodby now,particularly those pertaining to the linear dynamics, such as the degeneracy of azimuthalmodes in systemswith discrete ro-tational symmetry. The most prominent action of the nonlinearflame response is to destabilize thestanding-wave pattern sothata spinning az-imuthal mode is established at finite oscillation levels [10,11]. Comprehensive nonlinear analysis
[12] shows more complex phenomena, such as thesimultaneousexistenceof stablestandingand spinningwaves,recentlyobservedin theformof mode hysteresis [13]. Another phenomenon spe-cifictoannularchambersanddiscoveredrecently istheoccurrenceof anonlinearcouplingbetween axisymmetricandazimuthalmodes,givingriseto aslantedpatternof theperturbedflamesaround the combustion chamber circumference [8]. The present article is concerned with explaining the manifestation of the slanted mode based on a nonlinearthermoacousticmodel.
Fromamoregeneralpointofview,theslanted mode is the manifestation of nonlinear mode coupling that involves synchronization of two modeswithsimilarindividualresonance frequen-cies.Otheraspectsofsynchronizationhaverecently beenstudiedinthefieldofthermoacoustics[14,15], viz. forced synchronization and synchronization of differentfields. However,thepresentcase pre-ciselycorrespondstothescenariolabeledas ‘mu-tualsynchronizationofself-sustainedoscillators’in thestandardreference[16,Chap.4].
Fromexperimental and theoreticalstudies of thermoacousticinstabilitiesinlongitudinal config-urations,itappearsthatwhenevertwomodeswith differentresonancefrequenciesareunstableatthe sametime,theprevailingtendencyisthatonlyone modesurvivesandtheotherissuppressed[17],or aquasi-periodic solutionemerges, inwhichboth modes oscillate with different, generally incom-mensuratefrequencies[18,19].Sincesingle-burner configurations host only longitudinal modes in thelow-frequencyregime,twomodescannothave close resonance frequencies, unless the system features decoupled plenum and chamber modes
[20];therefore, synchronizationbetween different modesislesslikelytooccurinthesesystems.An annular chamber, however, has a significantly highermodaldensitysothatitmayindeedhappen thattwomodeshavesimilarresonancefrequencies, givingrisetosynchronization.
Ouranalysis isbasedon theoscillator model introducedbyNoirayetal.[11]forthe investiga-tionof theeffectof asymmetryonthestanding– spinningmodestructure.Thesamemodelwasused byGhirardoetal.[21]toexplainstandingmodes insymmetricsystemsthroughanonlinear mech-anism involving transverse velocity fluctuations. Thismodel isextendedhere toexaminethe cou-plingbetween anazimuthalandanaxisymmetric mode,correspondingtotheslantedoscillation pat-tern observed in previous experiments [8]. Since theazimuthaleigenspaceisspannedbytwomode shapes(sineandcosineorclockwiseand counter-clockwiserotating),theadditionofan axisymmet-ricmodeyieldsasetofthreeoscillatorequations describingtheevolutionofthemodesintime. Be-causethemodeshapesareorthogonal,thereisno linearcouplingbetweentheoscillators.Their inter-actionemergesonlythroughthecubictermandis thereforefullynonlinear.Theoscillatorequations andanassociatedslow-timesystemforthe ampli-tudesandphasesarederivedinSection 2. Com-prehensiveanalysisofthissystemiscarriedoutin
Section3.Section4discussestheanalyticalresults inviewofexperimentalobservations.
2. Modelequations
Theoscillatormodelwe considerissimilarto theoneproposedbyNoirayetal.[11].However, weincludeanadditionalaxisymmetricmodeand focusoncaseswheretheresonance frequencyof thisaxisymmetricmodeisclosetothatof the az-imuthalmode.Asinpreviouswork,westartfrom thewaveequationfortheacousticpressurewitha sourcetermassociatedwiththeunsteadyheat re-leaserateintheflames:
∂2p˜ ∂t˜2 +α˜ ∂p˜ ∂t˜ − c R 2∂ 2˜p ∂θ2 + ω 2 0p˜=(γ − 1) ∂q˜ ∂t˜. (1) Here, p˜is theacousticpressure,α a˜ damping coefficient,cthespeedofsound,Rthemeanradius oftheannulus,θ theangularcoordinate,γ the ra-tioofspecificheats,andq˜theheatreleaserate.The resonancefrequency of thefirst azimuthalmode isωa=c/R.Thetermω20p˜representsthe
axisym-metricmode,withresonancefrequencyω0.An
ax-isymmetricmodehasthesamephaseatthe circum-ferentiallocationsoftheflames;assuch,itcanbe alongitudinaloraHelmholtzmode.Theω2
0p˜term
canbe motivated bythefactthat applicationof theLaplaciantoanaxisymmetricmoderesultsina termproportionaltoω2
0p˜,withanadditionalfactor
Non-dimensionalvariablesareintroduced:t=
˜
tωa, p= p˜/(ρc2),q=q˜ρc2ωa/(γ − 1),α =α/ω˜ a, =ω0/ωa;ρ denotesthemeanfluiddensity.The
unsteadyheatreleaserateqistakentodependon the local pressure according to q(p)=β p− κ p3
[11];β andκ arepositiveconstants.Thisflame re-sponsemodelissimplebuthasbeenusedinmany recenttheoreticalstudies (e.g.,Refs.[11,21])asit encapsulatesthemostimportantnonlineareffect, viz.saturation.
Wenowassumeasolutionoftheform
p(θ,t)=ηc(t)cosθ +ηs(t)sinθ +η0(t). (2)
This corresponds to the expression used in Refs. [11,12,21], except the additional term η0,
which represents the axisymmetric mode. The cosθ and sinθ terms span the two-dimensional eigenspace of the degenerate azimuthal mode. For a spinning mode, theamplitudes of ηc and ηs areequal,andtheirphasedifferenceis ± π/2.
A standing wave has arbitrary amplitudes but a phasedifferencebetweenηcandηsof0orπ.
Theansatz(2)isnowintroducedintothewave
equation(1),andafterprojectionontothespatial basis{cosθ,sinθ,1},coupledoscillatorequations forηc,ηs,andη0areobtained:
¨ ηc/s+ηc/s= (β−α)η˙c/s− 3 4κ 3η2 c/s+η 2 s/c+4η 2 0 ˙ ηc/s +2ηc/sηs/cη˙s/c+8ηc/sη0η˙0 , (3a,b) ¨ η0+2η0= (β − α0)η˙0− 3κ (ηcη˙c+ηsη˙s)η0 +η2 c/2+η 2 s/2+η 2 0 ˙ η0 . (3c)
Here, (·)˙ denotes a derivative with respect to time, andthe damping coefficient for the ax-isymmetricmode,α0,has beenendowedwithan
additional subscript to indicate thatthis param-eter should be allowed to be generally different fromthosecorrespondingtotheazimuthalmode. The damping rate depends on the spatial struc-tureof themodeandtherefore maybe different for each oscillator. Conversely, β is a property of the flame response and thusidentical forall modes/oscillators.
Dampingandlinearflameresponse gainonly appearasdifferencesinEq.(3),correspondingto thelineargrowthratesofthemodes.Inthe follow-ing,thesegrowthratesaredenotedbyσa=β − α
fortheazimuthalmodeandσ0=β − α0forthe
ax-isymmetricmode.Sincewewishtostudythe inter-actionof azimuthalandaxisymmetricmodes,σa
andσ0areassumedpositive.
Thetermsinthesquarebracketsmultiplyingη˙c/s
inEq.(3a,b)andη˙0 in(3c)correspondto
nonlin-eardamping,whichwilllimittheamplitudegrowth of therespectivemodeatfiniteoscillationlevels. The term4η2
0 willthusreducethegrowthof the
azimuthalmode whilethe terms (η2
c+ηs2)/2 will
damptheaxisymmetricmode.Thisscenario pro-motes mode competition. Also note that in the absence of theaxisymmetric mode, with η0=0,
Eq.(3a,b)areidenticaltothesystemconsideredby Noirayetal.[11]inthestudyofapureazimuthal mode.
Foragivensetof parametersandinitial con-ditions,theoscillatorequations(3)canbesolved numerically, but this approach does not provide muchinsight.Wewillinsteadapplythemethodof averaging[22]tothesystemofcoupledoscillators, which allows drawing more general conclusions about the system dynamics. The solutions are assumedtobeoftheform
ηi=Ai(t)cos[t+φi(t)], i={c,s,0},
wheretheamplitudesAiandphasesφiareassumed
tobeslowlyvaryingfunctionsof time.Notethat theoscillationfrequencyisnotapriorifixedto1 (ωa in dimensionalvariables). A small deviation
from the resonance frequency of the uncoupled modecanbeaccommodatedintheslowlyvarying phasetermφ(t).
Followingstandardprocedures[22],the evolu-tionequationsfortheslowlyvaryingamplitudeand phasevariablesareobtainedbyaveragingthe con-tributionsoftheoscillatingquantitiesoverone pe-riod:
˙
Ai=−fisin(t+φi), φi˙ =−
1
Ai
ficos(t+φi),
(4) wherethefiaretherighthandsidesofEq.(3),and
theangledbracketsdenoteanaverageoverone pe-riodoftheuncoupledazimuthalmode.Notethata delayintheflameresponsemodelonlyaffectsthe lineargainbutnotthestructureoftheaveraged sys-tem[11].
There are four parameters in the oscillator system: the growth rates of the azimuthal and theaxisymmetric modes σa and σ0,theratio of
the oscillationfrequencies , and thesaturation coefficient κ from the nonlinear flame model. Twooftheseparameterscanberemovedthrough suitablerescaling.Tothiseffect,weintroduce
ˆ Ai=Ai κ σ0 1/2, σ =ˆ σa σ0 , =ˆ 2− 1 2σ0 , τ =tσ0.
Furthermore,the amplitudeandphaseevolution equations do not depend on the phasevariables independently but only on their differences. We therefore introduce phase-difference variables, definedasψi j=φi− φj.
Usingtheaveraging ansatz(4),theamplitude andphase-shift evolutionequationsarethen ob-tainedas
ˆ Ac/s= ˆ Ac/s 32 16σ − 3ˆ 3Aˆ2c/s+ 2Aˆ 2 s/c+ 8Aˆ 2 0 (5a,b) +Aˆ2 s/ccos2ψcs+4Aˆ20cos2ψc/s0 , ˆ A0= ˆ A0 32 16− 34(Aˆ2c+ ˆA 2 s+ ˆA 2 0) +2Aˆ2 ccos2ψc0+2Aˆ2scos2ψs0 , (5c) ψ cs = 3 32 ˆ A2 s + ˆA 2 c sin2ψcs +4Aˆ2 0(sin2ψc0− sin2ψs0) , (5d) ψ c0 =− ˆ + 3 32 ˆ A2 ssin2ψcs+4Aˆ20sin2ψc0 +2Aˆ2 csin2ψc0+2Aˆ2ssin2ψs0 . (5e)
Here,ψs0=ψc0− ψcs,and(·)denotesa
deriva-tive with respecttothescaled timeτ. Alsonote thatto leading order, thefirst term onthe right handsideofEq.(5e)correspondstothefrequency differencebetweentheazimuthalandthe axisym-metricmode.Forsmallamplitudes,thephasesof the axisymmetric and the azimuthal modes will thusdivergeatarateequaltothefrequency differ-encewhentheiruncoupledresonancefrequencies arenotidentical.However,thenonlineartermsin
(5e)may compensate the frequency difference to es-tablishafixedphasedifferencebetweenazimuthal andaxisymmetricmodes incaseof synchroniza-tion.Furthermore, it isevident from the system
(5)thatcouplingbetween themodes occursina purelynonlinearfashion.
In the following, we consider the system of nonlinearlycoupleddifferentialequations(5)that evolvethesystemstateinafive-dimensionalspace. Theadvantageofstudyingtheaveragedsystemis thatperiodicandquasi-periodicsolutionscanbe determinedmucheasiercomparedtotheoscillator
Eqs.(3).Periodicsolutionsofanykindare charac-terizedbyfixedpointsoftheaveragedsystem,i.e. [Aˆc,Aˆs,Aˆ0,ψcs,ψc0 ]=0, withatleast oneof the
amplitudesbeingnon-zero.Aquasi-periodic solu-tionwouldbecharacterizedbyatleastoneofthe azimuthalmodeamplitudesandtheaxisymmetric modeamplitudebeingnon-zeroandallofthe pre-viously mentioned state variables being constant exceptforψc0,whichwouldvarylinearlyintime,
driven by the frequency difference between the azimuthalandtheaxisymmetricmode.Stabilityof theindividualperiodicorquasi-periodicsolutions thatmayexistforcertainvaluesoftheparameters isalsomoreeasilydeterminedfromtheaveraged system.Oneonlyneedstoinspecttheeigenvalues of the Jacobian evaluated at the corresponding equilibriumstate.
Fig. 1.Comparison of numerical solutionsof the os-cillatorequations(3)andtheaveragedsystem(5).Top: modecoefficients ηi andmode amplitudesAi;the
lat-terareshownasgraylines.Bottom: phaseshiftsfrom thenumericalsolutionoftheaveragedsystem. =1.03,
σa=0.1,σ0=0.09,κ =0.2,correspondingtoσ ≈ 1ˆ .11
and ≈ 0ˆ .338.
3. Results,discussion,andfurtheranalysis
To assess the ability of the slow-time model
(5) to represent the dynamics of the original coupledoscillatorsystem(3),numericalsolutions based on both sets of equations are compared (Fig.1,topframe). Theamplitudesfromthe av-eragedsystemappearasenvelopesof the oscilla-torsolutions,anditisapparentthattheaveraged systemisaverygoodapproximationthatcaptures eventhenon-monotonicamplitudeevolution ac-curately. The parameter values for this case are
= 1.03,σa= 0.1,σ0= 0.09,κ = 0.2,withlinear
growthratesandsaturationcoefficient chosenas inRef.[11].Intermsof therescaledparameters, thiscorresponds toa growthrate ratioσ ≈ 1ˆ .11 andfrequencydetuning parameter ≈ 0ˆ .338.In theinitiallinearstage,allamplitudesgrow expo-nentiallybecauseofthepositivegrowthrates.After atransientstagewithnon-trivialamplitude evolu-tion,theaxisymmetricmodeeventuallydecaysto zeroandcosineandsineamplitudesattain identi-caloscillationlevels.Thephasebetweencosineand sinemodes,ψcs,settlesat−π/2(Fig.1,bottom),
whichshowsthatapurelyspinningmodeis estab-lished.Alsonotehowthephaseshiftbetween az-imuthalandaxisymmetricmodes,hererepresented throughψc0,increaseslinearlyatsmalloscillation
amplitudes,isnearlyconstantintheintermediate regimewhenallmodesfeatureappreciable ampli-tudes,andfinallydriftsoffwhentheaxisymmetric modehasvanished.Thisindicatesthatthesystem isalreadyclosetosynchronization.
Now the frequency ratio is decreased to
=1.01,correspondingtoafrequencydetuning parameter =ˆ 0.112 (Fig. 2). The solution of
~ ;§. 0.5
1
0î
-0.5 -1 ~ - - - - ~ - ~ - - ~ - - - - ~tJ
0~
50 100 150 200 250:
300l
non-dimensional time tFig.2. AsFig.1butwith =1.01,correspondingto ˆ
≈ 0.112.
theaveraged systemagaincorresponds wellwith that of the oscillator model. In contrast to the previouscasewithalargerfrequencydetuning,all modessurviveinthelong-timelimitandsettleon constant amplitudes (visible attimes larger than showninFig.2).Wenotethattheazimuthalmode isstanding,indicatedbyψcs=0(Fig.2,bottom).
Theamplituderatioofthecosineandsine compo-nentsdeterminestheorientationofthenodalline of the standing azimuthal mode. In the present axisymmetric setting, all orientationsareequally permissible,anditdependsontheinitialconditions whichisestablished.Furthermore,thephase differ-encebetweentheazimuthalandtheaxisymmetric mode,ψc0,settlesonafixedvalue,slightlylessthan π/2.Afixedphasedifferenceψc0 impliesthatthe
twomodes,whichhavedifferentlinearresonance frequencies,havebecomesynchronizedthroughthe nonlinearcoupling.Thisscenarioprecisely corre-spondstotheslantedmodedocumentedinRef.[8]. A phaseshift closetoπ/2 between azimuthal andaxisymmetricmodesisalsoobservedinthe av-eragedmodelandinexperimentaldata,tobe dis-cussedinthefollowing,andthereforeappearsto bearobustfeatureof thephenomenon. Further-more,thisphaseshiftisonlyinsignificantlyaffected by atime delay inthe flame response model, as wasverifiednumericallyonthebasisofthe oscilla-torsystem(3).Whileadelayintheflameresponse wouldgenerallyaffectthephaseshiftbetween pres-sureandheatreleaserate,thenonlinearcoupling ofthetwomodesoccursonlythroughthepressure atthelocationof theflames.Inthephase-locked state,bothmodesoscillateatthesamefrequency and,hence,acquirethesamephaseshiftfromthe flameresponse;thephaseshiftbetweenthemodes is therefore unaffected by the delay. Similarly, a morerealisticburnerimpedanceisnotexpectedto affectthisphaseshift.
In the following, we analyze the averaged system(5)to establishconditions inthereduced
parametersσ (growthˆ rateratio)and (frequencyˆ detuning),forwhichtheslantedmode,i.e.,the syn-chronizedoscillationbetweenanaxisymmetricand anazimuthalmode,existsandisstable.Sincethere isrotationalsymmetry,wecanassumewithoutloss ofgeneralitythatAs=0forastandingazimuthal
mode.TheamplitudeEqs.(5)thentaketheform ˆ Ac= ˆ Ac 32 16σ − 3ˆ 3Aˆ2c+ 4Aˆ 2 0(2+cos2ψc0) , (6a) ˆ A0= ˆ A0 32 16− 34Aˆ20+ 2Aˆ 2 c(2+cos2ψc0) , (6b) withphase-differenceequation
ψ c0=− ˆ + 3 16 ˆ A2 c+ 2Aˆ 2 0 sin2ψc0. (6c)
SinceAs=0,thephasedifferenceψcs is
irrele-vant.Foraslantedmode,therighthandsidesof
Eqs.(6)mustvanishfornon-zeroAˆc andAˆ0.We
firstdetermine conditionson andˆ σ forˆ which suchsolutionsexistandthenaddressthestability ofthesynchronizedstate.
From requiring that the right hand sides of
Eqs. (6) vanish, an equation only involving the phase-shiftvariableψc0canbeobtained:
2(2+cos2ψc0)2− 3 ˆ = 1/2+σ +ˆ (1+σ )ˆ cos2ψc0 sin2ψc0. (7)
This equation is quartic in cos2ψc0 and thus
may have four real solutions in any half-closed interval of length π. It appears that these so-lutions cannot be expressed in simple form for arbitrary andˆ σ .ˆ We therefore consider first the special casewhere theresonance frequencies of theazimuthalandtheaxisymmetricmodeare identical, i.e., =ˆ 0. Equation (7) then has the four solutions ψc0={0,π/2,a,−a}, where a is
given by a=arccos[−(σ +ˆ 1/2)/(σ +ˆ 1)], which areallmoduloπ.Whenthesesolutionsareused inrequiringthattherighthandsidesof (6a)and
(6b) vanish, explicit expressions for the squared amplitudes of thetwo modes are obtained. The solutions ψc0=±a lead to negative amplitude
squaresfortheaxisymmetricmodeandtherefore canbediscarded.Withtheothertwosolutionsof thephase-shiftequation(7),weobtain
ψc0=0: Aˆ2c= 16 45(3− ˆσ ), Aˆ 2 0= 4 15(2σ − 1)ˆ , (8) ψc0= π 2 : Aˆ 2 c= 16 3 (σ − 1)ˆ , Aˆ 2 0= 4 3(3− 2σ )ˆ . (9) These solutions correspond to phase coupled oscillationsof anazimuthalandanaxisymmetric
~
i.
0.5 ] 0 'à ~ -0.5 -1 ...__ ... _ _ ..__ ... _ _ ..._ _ __. _ _ _.I~
0~
50-
100~
150 200~
: :
250 300l
non-dimensional time tu~~-~-~-~-~~-~-~~ 1.6 <b 1.5 .g 1.4
e
~ 1.3 .c 1.2!
1.: 0.9 0.8 '---'---'----'----'---''----'----'----' -0.3 -0.2 -0.1 0 0.1 0.2 0.3frequency detuning 6
Fig. 3. Existence and stability domain (gray shaded) of the phase-coupled two-mode solution - the slanted mode - in parameter space. Stars indicate parame-ter combinations for which phase planes are shown in
Fig. 4. Diamond and square correspond to parameters of
Figs. 1 and 2, respectively.
mode, and it can be verified numerically that these are indeed equilibrium states of the full system given by Eqs. (5). Since the amplitude squares need to be positive, the conditions for the existence of a phase-coupled two-mode solution with phase
shift O and 7r/2 are 1/2 < 8 < 3 and 1 < 8 < 3/2, respectively. As we will show Jater in this section, the in-phase solution with
V/d)
=
0 is unstable so that the occurrence of a slanted mode is limited to the stricter requirement on the growth rate ratio.When the frequencies of the two modes are not identical, i.e.,
!!>
=I= 0, the phase-shift Eq. (7) can-not be solved in simple terrns. However, the rangeof growth rate ratios 8 that permits a stable
phase-coupled solution can be determined numerically as a fonction of the frequency detuning
6.,
as shownin Fig. 3. The effect of the frequency detuning is to reduce the range of growth rate ratios for which a slanted mode occurs. For
1
6.I
~ 0.273, nosta-ble phase-coupled two-mode solution exists for any growth rate ratio. For a non-dimensional growth
rate a0
=
0.2, this corresponds to a frequency dif-ference of about 5%. The stability domain of the slanted mode is symmetric with respect to!!>
be-cause the system (6) has the symmetry (!!>, 1/f,,,) ~(-!!>,
-
VJ,,,).
When!!>
«
1 is assumed, which isrea-sonable in view of Fig. 3, the effect of the fre-quency detuning on the phase shift can be deter-mined to Jeading order from Eq. (7) as 1/f"'
=
7r /2-6.
(modulo7r). Hence, the azimuthal and theax-isymmetric modes will always be approximately in
quadrature when they are phase coupled.
It is now instructive to consider the structure of the state space defined by the system (6) to under-stand the qualitative changes in the stability of the
synchronized two-mode solution. To this purpose, we plot the vector field defined by (6a) and (6b) cor-responding to different points in the parameter space (Fig. 4a). The system is three-dimensional,
and the 1/f"' coordinate can be thought of as
pointing out of the paper plane; however, we have
already shown that only the immediate vicinity of the planes
V/d)
=
0 andV/d)
=
7r /2 are of interest.Figure 4 illustrates the state space dynamics at
1/f"'
=
7r /2 for zero frequency detuning and a growth rate ratio of 8=
1.25, which is most favor-able for the existence of the slanted mode accordingto the stability map (a in Fig. 3). By construction,
the origin is a repeller; the single-mode limit cy-cles found on the axes, where one mode has zero amplitude, both correspond to saddles: they are
at-tracting in the direction of the non-zero mode, but repelling perpendicular to it. The in-quadrature
synchronized two-mode solution is a stable node,
globally attracting in the 1/f,,,
=
7r /2 slice of the state space. ln the slice corresponding to in-phasetwo-mode oscillations, 1/f"'
=
0 (Fig. 4b), the roles of the single-mode and the two-mode oscillations are reversed; the latter cannot occur as it isunsta-ble. Also note that the single-mode oscillations are attracting only in this plane, as the phase
equilib-rium at 1/f,,,
=
0 is unstable according to Eq. (6c). As the growth rate ratio 8 is decreased from 1.25, the stable node, corresponding to thesynchro-nized two-mode solution, successively approaches the saddle corresponding to the single-mode
ax-isymmetric oscillation. When 8 is decreased below
1 (see Fig. 3), the node collides with the saddle,
rendering the single-mode axisymmetric limit cycle stable (Fig. 4c). When 8 is increased beyond 3/2, the stable two-mode oscillation collides with the
single-mode azimuthal limit cycle, which is then the only stable solution (Fig. 4d). These latter effects are not
entirely surprising when one recalls that 8 is the
ra-tio of the growth rates of the azimuthal and the
ax-isymmetric mode.
Only the case with zero frequency detuning is
depicted in Fig. 4. However, the (Â~. Â
0)
vect or-field varies smoothly with 1/f ,,,, and the equilibriumphase shift changes only slightly with
!!>.
Qualita-tively, the dynamics therefore remains similar to that shown in Fig. 4 even when!!>
is non-zero, aslong as a stable synchronized two-mode solution exists
(
l
!!i
l
;§
0.273, Fig. 3). Whenl
!!il
increases be-yond this value, the synchronized two-modesolu-tion only exists as an unstable saddle, as in Fig. 4b,
and the only stable solutions are single-mode limit cycles.
4. Modeling results and experimental observations
The analysis in the previous section showed that
the slanted mode [8] arises from a synchronization
process between unstable azimuthal and
axisym-metric modes. Furtherrnore, this synchronization
is only possible when the individual resonance
fre-quencies of the modes are close and when their growth rates are similar. This is consistent with
oc-Fig. 4. Phase planes for the amplitude dynamics of the reduced standing-azimuthal-axisymmetric system, Eqs. (6a}-(6b). a)â = 1.25, t,. = 0, iftc0 = :ir/2; b)â = 1.25, t,. = 0, iftc0 = 0;c)â = 0.9, t,. = 0, iftc0 = ,r/2;d)â = 1.6, t,. = 0, iftc0 = ,r/2.
The amplitude range is Âc = 0 ... 1.6 for a)-c) and Âc = 0 ... 1. 7 for d). Âo ranges from O to 1.6 in ail frames. Filled
circles represent stable equilibrium solutions, open circles unstable ones. Background color corresponds to the norm of
the amplitude rate of change, ((Â~)2
+
(Â0
)2)112; colorscaJe saturates at value 1.
curs only in a very narrow region in the operating
space (see Fig. 4 in Ref. [13D.
lt would be interesting to test the predicted dependence of the slanted mode solution on the frequency detuning and the growth rate ratio,
as illustrated in Fig. 3. This is unfortunately not
possible because (i) the frequency detuning and
the growth rates cannot be varied independently in the experiment, and (ii) these parameters
can-not be accurately measured. However, the phase
shift between the azimuthal and the axisy=etric mode, 1/fc0, which is predicted to be close to 1e/2
from the analysis, can be retrieved from available
experimental data. To this purpose, we analyze
high-speed images of the flames' light emission
acquired during conditions corresponding to the
slanted mode (same data set as was used in Ref. [8]). The images were acquired with a Photron Fastcam APXii at a framerate of 12,000 images per second. A subset of 1000 images was analyzed by means of
proper orthogonal decomposition (POO). Increas-ing the number of images did not noticeably affect
the results presented in the following.
POO decomposes snapshots of a space and time
dependent observable into orthogonal modes and
ranks them according to their fluctuation energy
[23]. The fluctuation energy of the first 100 modes relative to the total fluctuation energy is shown in
Fig. 5. l'wo modes are clearly dominant, ail others
contributing with Jess than 1% to the total
fluctu-ation energy. The first two POO modes are shown
in Fig. 5 as insets. The POO mode with the largest fluctuation energy can be identified as axisymme
t-ric thermoacoustic mode, all flames featuring the
same sign of the heat release rate perturbation. ln contrast, the POO mode with the second highest fluctuation energy is a first-order azimuthal mode,
with positive and negative heat release rate
fluc-tuations varying sinusoidally around the circum-ference. This corresponds to a standing azimuthal
mode; a rotating azimuthal mode would be
repre-sented by two POO modes rotated with respect to each other by an angle of 1e/2.
~
:.
0 -...
-
.
-~max~
:in
0 ~.
.
_ ... ... . ..
.
.
.
.
.
.
.
0 0.
1!
.. •
•• ·-
•
.
:•
·
-
::,i
---1 10• modenumber..
Fig. 5. Relative fluctuation energy of the first 100 POO modes obtained from high-speed imaging of the flames'
light emission in the MICCA combustor when the slanted
mode is observed. A Jong-exposure photo is shown in the
inset to the Jower Jeft. The first two POO modes are shown
as insets to the upper right. The light that can be seen
ad-jacent to the actual, disk-shaped flames stems from light
reflections on the quartz combustor walls.
0.1 ~ - ~ - ~ - ~ - ~ - ~ - ~ - - ~
- axisymmetric - azimuthal
] 0.05
l
o .g ~ -0.05 -0.1 ' - - - ~ - ~ - ~ - ~ - ~ - ~ - - . . . _ , 0 2 4 6 8 10 12 14 time (ms)Fig. 6. Amplitudes corresponding to the axisymmetric and the azimuthaJ modes, obtained from POO of the high-speed images.
The phase relation of the two dominant POO modes can be deduced from their associated time
coefficients. These indeed exhibit a phase shift very
close to 1e/2, showing that the two modes are in quadrature (Fig. 6). The precise value of the mean
phase shift is obtained from the cross spectrum of the two time coefficients; this evaluates to 1.595 rad at the dominant frequency (450 Hz), whichdeviatesfromexactquadraturebylessthan 2%. This result corroborates the analysis in the preceding section,which predicted a quadrature relation whenever an azimuthal and an axisym-metricmodebecomesynchronizedinatwo-mode oscillation. Theinstantaneous phaseshiftcan be determinedfromtheassociatedanalyticsignalsvia the Hilbert transform. This quantity shows very littledeviationfromthemeanvalue,witha stan-darddeviationof lessthan0.03rad,highlighting propersynchronizationbetweenthetwomodes.
5. Conclusion
Nonlinear couplingof thermoacoustic modes inannularcombustorswasinvestigated.A synchro-nized two-mode limit cycle involving a standing azimuthal and an axisymmetric oscillation, as recentlyobservedinanannularmodelcombustor
[8], was analyzed. This phase-coupled two-mode oscillation occurs provided that the individual resonance frequencies of the two modes and theirgrowthratesaresimilar.Thesefindingswere obtained bystudyingareduced systemthatonly allowsforastandingazimuthalmodeinaddition to the axisymmetric mode. However, it can be shown that a synchronized oscillation involving aspinningazimuthalmodeandanaxisymmetric modedoesnotexist.
Through suitable rescaling, the dependence of the system dynamics on the cubic coefficient
κ, which controls the saturation of the flame response,isremoved(providedκ remainspositive). Theexistenceoftheslantedmodesolutionisthus supportedby alargeclassof flame modelswith saturation nonlinearity. The specific value of κ affectsthesolutiononaquantitativelevel(the os-cillationamplitude),buttheanalysisshowedthat thequalitativedynamics,i.e.,single-mode oscilla-tions or phase-locked two-mode oscillations,are unaffected.Thequalitativedynamicsdependonly onthetwonon-dimensionalparametersσ andˆ ,ˆ correspondingtogrowthrateratioandfrequency detuning,respectively.Wecanthenexpectthatthis nonlinearlysynchronizedtwo-modeoscillationisa rathergenericfeatureofthermoacousticinstability inannularcombustorsthatmayappearwhenthe
resonancefrequenciesandthegrowthratesof an azimuthalandanaxisymmetricmodeareclose.
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