Nonlinear thermoacoustic mode synchronization in annular combustors

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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

c

a_{DepartmentofEnergyandProcessEngineering,Norwegian UniversityofScienceandTechnology,Trondheim7491,} Norway

b_{InstitutfürStrömungsmechanikundTechnischeAkustik,TechnischeUniversitätBerlin,Berlin10623,Germany} c_{LaboratoireEM2C,CNRS,CentraleSupélec,Université Paris-Saclay,Gif-sur-Yvettecedex92292,France} d_{InstitutMécaniquedesFluidesdeToulouse,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

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-tudesandphasesarederivedin Section 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:

∂ 2_{p ˜} ∂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,c thespeedofsound,R themeanradius

oftheannulus,θ theangularcoordinate,γ the

ra-tioofspecificheats,andq ˜theheatreleaserate.The

resonancefrequency of thefirst azimuthalmode

is ω a= c/R .Theterm ω 20p ˜representsthe

axisym-metricmode,withresonancefrequencyω 0.An

ax-isymmetricmodehasthesamephaseatthe

circum-ferentiallocationsoftheflames;assuch,itcanbe

alongitudinaloraHelmholtzmode.Theω 2

0p ˜term

canbe motivated bythefactthat applicationof

theLaplaciantoanaxisymmetricmoderesultsina

termproportionaltoω 2

0p ˜, withanadditionalfactor

Non-dimensionalvariablesareintroduced: t =

˜

t ω a, p = p ˜/ (ρc 2), q =q ˜ρc 2ω a/ (γ − 1) , α = α/ω ˜ a,  =ω0/ωa;ρ denotesthemeanfluiddensity.The

unsteadyheatreleaserate q istakentodependon

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

phasedifferencebetweenηcandηsof0or π.

Theansatz (2)isnowintroducedintothewave

equation(1),andafterprojectionontothespatial

basis{cosθ, sinθ, 1},coupledoscillatorequations

forηc,ηs,and η0areobtained:

¨ 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

appearasdifferencesin Eq.(3),correspondingto

thelineargrowthratesofthemodes.Inthe

follow-ing,thesegrowthratesaredenotedby σa= β − α

fortheazimuthalmodeandσ0= β − α0forthe

ax-isymmetricmode.Sincewewishtostudythe

inter-actionof azimuthalandaxisymmetricmodes, σa

andσ0areassumedpositive.

Thetermsinthesquarebracketsmultiplyingη˙c/s

in Eq.(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,theoscillator equations(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= A i(t)cos[t +φi(t)], i ={c,s,0},

wheretheamplitudesA iandphasesφiareassumed

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: ˙ A i=−f isin(t+φi), φ˙i=− 1 A i f icos(t+φi), (4)

wherethe f iaretherighthandsidesof Eq.(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 A i=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,

definedasψ i j= φi− φj.

Usingtheaveraging ansatz (4),theamplitude

andphase-shift evolutionequationsarethen

ˆ A c/s= ˆ A c/s 32 16σ − 3ˆ  3A ˆ2c/s+ 2A ˆ 2 s/c+ 8A ˆ 2 0 (5a,b) +_A ˆ2 s/ccos2ψcs+4A ˆ20cos2ψc/s0  , ˆ A 0= ˆ A 0 32 16− 34(A ˆ2c+ ˆA 2 s+ ˆA 2 0) +2A ˆ2 ccos2ψ c0+2A ˆ2scos2ψ s0  , (5c) ψ  cs = 3 32 _ˆ A 2 s + ˆA 2 c  sin2ψ cs +4A ˆ2 0(sin2ψ c0− sin2ψ s0)  , (5d) ψ  c0 =− ˆ + 3 32  _ˆ A 2 ssin2ψ cs+4A ˆ20sin2ψ c0 +2A ˆ2 csin2ψ c0+2A ˆ2ssin2ψ s0  . (5e)

Here,ψs0=ψc0− ψcs,and(·)denotesa

deriva-tive with respecttothescaled time τ. Alsonote

thatto leading order, thefirst term onthe right

handsideof Eq.(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

nonlinearlycoupleddifferential equations(5)that

evolvethesystemstateinafive-dimensionalspace.

Theadvantageofstudyingtheaveragedsystemis

thatperiodicandquasi-periodicsolutionscanbe

determinedmucheasiercomparedtotheoscillator

Eqs.(3).Periodicsolutionsofanykindare

charac-terizedbyfixedpointsoftheaveragedsystem,i.e.

[A ˆc,A ˆs,A ˆ0,ψcs,ψc0 ]=0, withatleast oneof the

amplitudesbeingnon-zero.Aquasi-periodic

solu-tionwouldbecharacterizedbyatleastoneofthe

azimuthalmodeamplitudesandtheaxisymmetric

modeamplitudebeingnon-zeroandallofthe

pre-viously mentioned state variables being constant

exceptfor ψ c0,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-cillator equations(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_{ ≈ 0}ˆ _.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 ≈ 1 ˆ . 11

andfrequencydetuning parameter ≈ 0 ˆ . 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

through ψ c0,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

0

î

-0.5 -1 ~ - - - - ~ - ~ - - ~ - - - - ~

tJ

0

~

50 100 150 200 250

:

300

l

non-dimensional time t

Fig.2. As Fig.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,indicatedby ψ cs=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,ψ c0,settlesonafixedvalue,slightlylessthan

π/2. Afixedphasedifference ψ c0 impliesthatthe

twomodes,whichhavedifferentlinearresonance

frequencies,havebecomesynchronizedthroughthe

nonlinearcoupling.Thisscenarioprecisely

corre-spondstotheslantedmodedocumentedinRef.[8].

A phaseshift closeto π/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 (growth ˆ rateratio)and (frequencyˆ

detuning),forwhichtheslantedmode,i.e.,the

syn-chronizedoscillationbetweenanaxisymmetricand

anazimuthalmode,existsandisstable.Sincethere

isrotationalsymmetry,wecanassumewithoutloss

ofgeneralitythat A s=0forastandingazimuthal

mode.Theamplitude Eqs.(5)thentaketheform

ˆ A c= ˆ A c 32 16σ − 3ˆ 3A ˆ2c+ 4A ˆ 2 0(2+cos2ψc0)  , (6a) ˆ A 0= ˆ A 0 32 16− 34A ˆ20+ 2A ˆ 2 c(2+cos2ψc0)  , (6b)

withphase-differenceequation

ψ  c0=− ˆ + 3 16  _ˆ A 2 c+ 2A ˆ 2 0  sin2ψ c0. (6c)

Since A s=0, thephasedifference ψ cs is

irrele-vant.Foraslantedmode,therighthandsidesof

Eqs.(6)mustvanishfornon-zeroA ˆc andA ˆ0.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-shiftvariableψ c0canbeobtained:

2(2+cos2ψ c0)2− 3  _ˆ  = 1/ 2+σ + ˆ (1+σ ) ˆ cos2ψ c0  sin2ψ c0. (7)

This equation is quartic in cos2ψ c0 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

areallmodulo π. Whenthesesolutionsareused

inrequiringthattherighthandsidesof (6a)and

(6b) vanish, explicit expressions for the squared

amplitudes of thetwo modes are obtained. The

solutions ψ c0=±a lead to negative amplitude

squaresfortheaxisymmetricmodeandtherefore

canbediscarded.Withtheothertwosolutionsof

thephase-shift equation(7),weobtain

ψ c0=0: A ˆ2c= 16 45(3− ˆσ ) , A ˆ 2 0= 4 15(2σ − 1) ˆ , (8) ψc0= π 2 : A ˆ 2 c= 16 3 (σ − 1)ˆ , A ˆ 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}

~

: :

_{250 300}

l

non-dimensional time t

u~~-~-~-~-~~-~-~~ 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 0.3

frequency 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 range

of growth rate ratios 8 that permits a stable

phase-coupled solution can be determined numerically as a fonction of the frequency detuning

6.,

as shown

in 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, no

sta-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 is rea-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 the

ax-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 and

V/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 according

to 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-phase

two-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 is

unsta-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 the

synchro-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 equilibrium

phase shift changes only slightly with

!!>.

Qualita-tively, the dynamics therefore remains similar to that shown in Fig. 4 even when

!!>

is non-zero, as

long as a stable synchronized two-mode solution exists

(

l

!!i

l

;§

0.273, Fig. 3). When

l

!!i

l

increases be-yond this value, the synchronized two-mode

solu-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

₊

_(Â

0

)2)1

12; 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 axisymmet-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 remains positive).

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 and ˆ _,ˆ

correspondingtogrowthrateratioandfrequency

detuning,respectively.Wecanthenexpectthatthis

nonlinearlysynchronizedtwo-modeoscillationisa

rathergenericfeatureofthermoacousticinstability

inannularcombustorsthatmayappearwhenthe

resonancefrequenciesandthegrowthratesof an

azimuthalandanaxisymmetricmodeareclose.

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