Flame Describing Function analysis of spinning and standing modes in an annular combustor and comparison with experiments

O

pen

A

rchive

T

OULOUSE

A

rchive

O

uverte (

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)

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To link to this article : DOI:10.1016/j.combustflame.2017.05.021

URL :

https://doi.org/10.1016/j.combustflame.2017.05.021

To cite this version : Laera, Davide and Schuller, Thierry

and

Prieur, Kevin and Durox, Daniel and Camporeale, Sergio M. and

Candel, Sébastien Flame Describing Function analysis of spinning

and standing modes in an annular combustor and comparison with

experiments. (2017) Combustion and Flame, vol. 184. pp. 136-152.

ISSN 0010-2180

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Flame

Describing

Function

analysis

of

spinning

and

standing

modes

in

an

annular

combustor

and

comparison

with

experiments

Davide

Laera

a,∗

_,

_Thierry

_Schuller

b,d

_,

_Kevin

_Prieur

b,c

_,

_Daniel

_Durox

b

_,

_Sergio

_M.

_Camporeale

a

_,

Sébastien

Candel

b

a Politecnico di Bari, 70125 Via Re David, 200, Bari (Ba), Italy

b EM2C Laboratory, CNRS, CentraleSupélec, Université Paris Saclay, 92295 Châtenay-Malabry cedex, France c Safran Tech, E&P, Châteaufort, CS 80112, 78772 Magny-Les-Hameaux, France

d Institut de Mecanique des Fluides de Toulouse (IMFT), Universite de Toulouse, CNRS, INPT, UPS, Toulouse, France

Keywords: Annular systems Combustion instabilities Spinning mode Standing mode Helmholtz solver Flame Describing Function

a

b

s

t

r

a

c

t

This article reports a numerical analysis of combustion instabilitiescoupled by a spinning mode or a standing mode in an annular combustor. The method combines an iterative algorithm involving a Helmholtz solver with the Flame Describing Function (FDF) framework. Thisis applied to azimuthal acousticcouplingwithcombustiondynamicsandisusedtoperformaweaklynonlinearstabilityanalysis yieldingthesystemresponsetrajectoryinthefrequency-growthrateplaneuntilalimitcycleconditionis reached.Twoscenariosformodetypeselectionaretentativelyproposed.Thefirstisbasedonananalysis ofthefrequencygrowthratetrajectoriesofthesystemfordifferentinitialsolutions.Thesecond consid-ersthestability ofthesolutionsatlimitcycle.It isconcludedthat acriterioncombiningthestability analysisatthelimitcyclewiththetrajectoryanalysismightbestdefinethemodetypeatthelimit cy-cle.SimulationsarecomparedwithexperimentscarriedoutontheMICCAtestfacilityequippedwith16 matrixburners.EachburnerresponseisrepresentedbymeansofaglobalFDFanditisconsideredthat thespacingbetweenburnersissuchthatcouplingwiththemodetakesplacewithoutmutual interac-tionsbetweenadjacent burningregions.Dependingonthe natureofthe modebeingconsidered,two hypothesesaremadeforthe FDFsoftheburners.Wheninstabilitiesarecoupledbyaspinningmode, eachburnerfeaturesthesamevelocityfluctuationlevelimplyingthatthe complexFDFvalues arethe sameforallburners.Incaseofastandingmode,thesixteenburnersfeaturedifferentvelocity fluctua-tionamplitudesdependingontheirrelativepositionwithrespecttothepressurenodalline.Simulations retrieve thespinningorstandingnatureofthe self-sustainedmodethat wereidentifiedinthe exper-iments bothintheplenum andinthe combustion chamber.The frequencyand amplitudeofvelocity fluctuationspredicted atlimitcycleareused toreconstructtime resolvedpressure fluctuationsinthe plenumand chamberandheatreleaseratefluctuationsattwolocations.Forthe pressurefluctuations, theanalysisprovidesasuitableestimateofthelimitcycleoscillationandsuitablyretrievesexperimental datarecordedintheMICCAsetupandinparticularreflectsthedifferenceinamplitudelevelsobserved inthesetwocavities.Differencesinmeasuredandpredictedamplitudesappearfortheheatreleaserate fluctuations.TheiramplitudeisfoundtobedirectlylinkedtotherapidchangeintheFDFgainasthe velocityfluctuationlevelreacheslargeamplitudescorrespondingtothelimitcycle,underlyingtheneed ofFDFinformationathighmodulationamplitudes.

1. Introduction

Many recentstudies focusoncombustion instabilitiescoupled by azimuthal modesin annularsystems.There are yet few com-parisonsbetweenpredictionsandwellcontrolledexperiments.The presentinvestigationaims atfilling thisgapby developinga

nu-∗ _{Corresponding author.}

E-mail addresses: davide.laera@poliba.it , davidelaera@gmail.com (D. Laera).

mericalprocedurecombiningtheFlameDescribingFunction(FDF) frameworkwithaHelmholtzsolvertoanalyzeazimuthal instabili-ties:

• Specificiterativealgorithms aredevelopedtosimulatethe dy-namicsofspinningandstandingmodeswithintheFDF frame-work.

• Thisis testedfirst ona systemrepresented by an idealflame responseandresultsofrecentanalyticalinvestigationsarefully retrievedvalidatingthismethodology.

• Twodifferentoperatingconditionsareconsidered,one leading toaspinninglimitcycleandanotheroneleadingtoastanding limitcycle.

• This framework is then used to calculate the limit cycles of standingandspinningsolutionsandtocomparethecalculated oscillationwithmeasurementson alaboratoryscale test facil-ity“MICCA” developedatEM2Claboratory.Theamplitudesand phaserelationshipsofpressureintheplenumandchamberand theheatreleaseratesignalsarecomparedfortwodifferent op-eratingconditions.

• Finally, two scenarios are tentatively proposed to explain the mode type selection. The first considers that the frequency andgrowth ratetrajectoriesof initially spinningand standing modes determine the solution at the limit cycle. The second suggestedby areviewerconsidersthatitisthestabilityofthe limitcyclewhichdeterminestheobservedoscillation.

At thispoint, itis worth briefly reviewingsome recent inves-tigations of instabilities in annular devices. Combustion instabil-ities coupled by azimuthal modes are often studied by theoreti-cal ornumericalmeans.Onthenumericallevelonefindsa grow-ingnumberofmassivelyparallellargeeddysimulationsofannular chamberswithmultipleburners [1–3].Thesecalculationsretrieve features ofazimuthally coupled combustioninstabilities observed in experiments in engine-like conditions [4]. However, the com-plexity ofthesituationprecludes direct comparisonbetween cal-culationsandobservations.

One difficulty inperforming suchcomparisons liesinthe def-initionofthe flamemodel.The firstinvestigations ofcombustion instabilitiescoupledtoazimuthalmodeswereperformedby com-bining a low-orderacousticmodel ofan annularcombustor with atime delayedn₋

τ

flameresponse (seeforexample [5,6]).This kindofmodelisalsoassumedintherecentanalyticalstudies de-velopedin [7,8]toanalyzethelinearstabilityofannularchambers fed by an annular plenum with multiple discrete burners. Both spinningandstandingmodesarepredicteddependingonthe cir-cumferential symmetryofthesystem.Circumferentialinstabilities of industrial combustors were analyzed in [9,10] by means of a Helmholtz solver approach. In these studies, the flame response was modeled by a n−

τ

description with parameters retrieved fromCFDcalculationsofthesteadycombustionprocess.These pre-viousstudiescarriedoutwithlineartoolscouldnotaccountfor fi-niteamplitudeeffectsthatdeterminetheoscillationfrequencyand levelatthelimitcycle.Thesefeatureswereconsideredforexample in [11,12] bycombiningdifferentnumericalstrategiesforacoustic propagationwithanonlinearflamemodelinthetimedomain. Nu-mericalcontrolstrategiesforannularconfigurationsfeaturing spin-ninglimitcyclesweredevelopedin [13,14].Spinningandstanding modeswereobservedin [11]dependingonwhetherthemeanflow velocitywasneglectedorconsideredinthesimulations.Fora cir-cumferentialinstabilityinanaxisymmetricgeometry,thespinning waveformwasalwayspreferredtothestandingmode inthe sim-ulations carried out in [5].However, no comparisonwith experi-mentswerereportedintheseworks.

On a theoretical level, criteria for appearance of spinning or standingmodeshavebeenderivedbyconsideringthedynamicsof azimuthal modescoupled by a nonlinear flame model expressed in terms of pressure perturbations alone. One of the first analy-siswascarriedoutin [15]withasaturation functionlinkingheat release andpressure fluctuations. In thisframework, the dynam-ics of azimuthal modes isdescribed by two harmonic oscillators whichare nonlinearlycoupled.Thestabilityofstandingand trav-elingwavesatlimitcyclecanthenbeassessed.Afurther simplifi-cationwaslater introducedby assumingthat thesystembehaves likeaVanderPoloscillator [16–18].Resultsindicatethatthe spin-ningorstandingnatureoftheunstablemode originatesfromthe

nonlinearityandnon-uniformityoftheflameresponseandcanbe influencedbydifferentfactors,liketransversevelocityfluctuations

[19,20]orturbulence,whichcanstochasticallydisturbthelimit cy-cleamplitudes.Thisnonlinearflamemodelwasusedforexample in [21]toreproducethedynamicalbehaviorobservedinareal en-gine.Anothercomparisonispresentedin [22]betweennumerical andexperimental growth rates calculated by means of a system identificationtechnique, butthe oscillationlevels ofthe different pressuresignalsarenotshown.

Recently,Ghirardoetal. [23,24]managedtointroduceintheir time domainmodela morereliable FDF,linkingheat releaseand pressuredisturbancesbyatime-invariantnonlinearoperator. Crite-riaforself-sustainedthermo-acousticinstabilitiescoupledby spin-ningandstandingmodeswerethenderivedbyexaminingthe sta-bility of the analytical solutions at limit cycles. This framework was then tested with experimental data from Bourgouin et al.

[25]whereastablespinningmodeisobservedatlimitcycle.Their analysisconfirmedthatfortheoperatingconditionexplored,there wasastablespinningsolutionandthat standingsolutions,ifthey existed, were unstable. In all of these previous studies there are nodirectcomparisonsbetweenpredictionsandmeasurementsfor pressureandheatreleaserateoscillationsandonlylimited valida-tionsofmodelpredictionsfordifferentoperatingconditions.

Thedifficulties encounteredinthesevarious investigationsare compounded by the presence of multiple flames which respond collectivelyover a wide frequencyrange, andby the modal den-sityintheannulargeometrywhenthesizeofthesystemislarge like in industrial gas turbine combustors. One possible simplifi-cationconsists inconsidering that the heat releasefromthe dif-ferentburners isuniformly distributedover the circumferenceof the annular chamber. Following this approach, Bourgouin et al.

[25] developed an analytical one-dimensional framework to rep-resent thedynamics ofthe laboratory scaleMICCA annular com-bustor. Assuminga simplified flameresponse,the spinning insta-bility recorded during experiments was reproduced in terms of frequencyand amplitudeof velocity fluctuations atthe limit cy-cle.Atheoreticalinterpretationwasalsoproposedfortheangular shiftobservedinthe experimentsbetweenthenodallines inthe plenumandinthechamber.However,thisanalysiswascarriedout fora fixed frequency andfixed oscillation levelcorresponding to thevaluesobservedintheexperimentatthelimitcycle.

On the experimental level there are relatively few data sets corresponding to instrumented conditions that can be used to benchmark models and simulations. Most of the measurements performed on real systems consist of unsteady pressure signals withnoaccesstotheflamedynamics [9,21,26].Bothspinningand standingmodepatternswereobservedinthelaboratoryscale an-nulardeviceequippedwithlowswirlinjectorsthatwasdeveloped intheengineeringdepartmentofCambridgeUniversity [4,27].This setupallows heatreleaseratemeasurements andflamedynamics analysisthroughoptical windows,buttheflametransferfunction wasnotdeterminedandpressuresignalswerenotrecordedinthe combustionchamber.

This article is organized asfollows. A novel procedure is de-rivedin Section 2that combinesa Helmholtzsolverwithsixteen independentFDFs. Thisis usedto determinethe limit cycle con-ditionsbymeansofaweaklynonlinearstabilityanalysis. Depend-ingonthenatureofthemodebeingconsidered,theFDFsare as-sumedtooperateatequaloratdifferentvelocityfluctuation lev-elsfortheexaminationofthedynamicsofspinningandstanding modes, respectively.This numericalprocedure isvalidatedin Ap-pendixAinan idealizedgeometricalconfigurationwitha simpli-fied flamemodel by retrieving the amplitudeand stability prop-ertiesof the theoretical limit cycles [16]. Section 3describes the MICCAcombustorexperimentwith16matrixlaminarinjectorsand thenumerical framework used forthe stabilityanalysis.The FDF

determined for one of the matrix injectors is presented in

Section 4 for the first operating condition leading to a spinning modelimitcycleintheMICCAcombustor.Thesametypeof analy-sisisrepeatedin Section5forthesecondoperatingpointleading toastandingmodelimitcycle.Thefrequencyandamplitudeof ve-locityfluctuationspredictedatlimitcycleareusedto reconstruct the pressure oscillations in the plenumand in the chamber and theheatreleaseratefluctuationsignal.Thesesignalsarethen com-paredwithmicrophonerecordsandphotomultipliermeasurements attwodifferentflamelocations. Amodetypeselectionanalysisis presentedin Section 6. Forthe two operating points, frequency-growthratetrajectoriesarecalculatedandthestabilityproperties ofthesimulatedlimitcyclesarediscussed.Theseanalysesare ten-tativelyusedtodeterminethelimitcyclestructure.

2. Weaklynonlinearstabilitynumericalanalysis

The methodology used to assess the stability of the annular combustoracousticmodes anddeterminethe limit cyclestate of thesystemfollowsthatdevelopedfortheFDFbasedweakly non-linearstabilityanalysisofsingleburnersetups [28–31].Inthetime domainonehastoconsiderawaveequationincludingadamping termdefinedbya firstordertimederivative ofthepressure fluc-tuationspmultipliedbyadampingrate

δ

(s−1) [32]andasource termrepresentingeffectsoftheunsteady heatreleaserate distur-bancesq˙:

∂

2_p

∂

t2 +4

πδ ∂

p

∂

t −

∇

·



c2

∇

p



=

(

γ

− 1

)

∂

_∂

q_t˙, (1)

whereithasbeenassumedthatthemeanpressurepisessentially constantsothat

ρ

c2isalsoconstant.Assumingthatallfluctuations areharmonicx = xˆexp

(

−i

ω

t

)

,oneobtainsthefollowingequation inthefrequencydomain:

ω

2 c2 pˆ+i

ω

4

πδ

c2 pˆ+

ρ∇

·



₁

ρ ∇

pˆ



=i

γ

− 1 c2

ω

ˆ_˙ q, (2)

where

ω

denotes thecomplexangular frequency.The mean den-sity

ρ

, speed of sound c and specific heat ratio

γ

distributions arespecified.Inthisfrequencydomain,theanalysisiscarriedout byusingthefiniteelement method(FEM)basedonthe commer-cial software COMSOL Multiphysics. This code solves the classical Helmholtzequation in which heat releaserate fluctuationsqˆ˙ are treatedaspressuresources.

Anonlineardescriptionoftheinteractionbetweencombustion andacoustics isrequiredtocapturethe limitcyclesofa thermo-acousticsystem.Iftheflameiscompactwithrespecttothe wave-length ofthe unstable mode, the dynamicsof the flame maybe representedintermsof aglobalFDFF [30],where theFDFgain andphaselag arefunction oftheamplitudeoftheincoming per-turbation [28,33]. In the frequency domain _{F links} relative heat releaseratefluctuationsQˆ˙/Q˙ torelativevelocityfluctuations

|

uˆ/u

|

measuredatareferencepointofthesystem.TheFDFisacomplex functionexpressedintermsofagainGandphase

ϕ

asfollows:

F

(

ω

r,

|

u/u

|

)

= ˆ_˙ Q

(

ω

r,

|

u/u

|

)

/Q˙

|

uˆ/u

|

=G

(

ω

r,

|

u/u

|

)

exp



i

ϕ

(

ω

r,

|

u/u

|

)



, (3)

where

|

u_/u

|

standsfortherelativevelocityfluctuationlevel,with u the root-mean-square ofthe velocity signalstaken at the ref-erenceposition in the injector unit j and

ω

r corresponds to the realpartofthecomplexfrequency

ω

.Aweaklynonlinearapproach is used to couple Eq. (3) with Eq. (2) retrieving the solution of thenonlinearproblemasaperturbationofalinearproblem. This

isachievedby linearizingthe FDFby fixinga velocity fluctuation level

|

u/u

|

. The finite element discretization of this set of lin-earized equations along with the boundary conditions results in thefollowingeigenvalueproblem [34,35]:

[A]P+

ω

[B

(

ω

)

]P+

ω

2_[C]P=[D

₍

_ω

₎

_{]P, (4)} whereP isthe pressureeigenmodes vector. Thematrices[A]and [C] contain coefficients originatingfrom the discretization of the Helmholtz equation, [B(

ω

)] is the matrixof the boundary condi-tions andof the dampingand[D(

ω

)] representsthe source term duetotheunsteadyheatrelease.Withtheintroductionoftheheat release rate the eigenvalue problem defined by Eq. (4) becomes nonlinearandissolvedwithaniterativealgorithm.Atthekth iter-ation Eq.(4)isfirstreducedtoalineareigenvalueproblemaround aspecificfrequency

k:

(

[A]+

k[B

(

k

)

]− [D

(

k

)

]

)

P+

ω

k2[C]P=0, (5) where

k=

ω

k−1 isthepreviousiterationresult.Thesoftwareuses theARPACKnumericalroutineforlarge-scaleeigenvalueproblems. This is based on a variant of the Arnoldi algorithm, called the implicitrestartedArnoldi method [36].This procedure isiterated until the error defined by



=

|

ω

k−

k

|

becomes lower than a specific value, typically 10−6. Once convergence is achieved, the real part of

ω

yields the oscillation frequency, f=-(

ω

)/2

π

Hz, while the imaginary part of

ω

corresponds to the growth rate

α

= 

(

ω

)

/2

π

s−1thatallowstheidentificationofunstablemodes: p_∝exp

(

2

πα

t_{− i}2

π

ft

)

.If

α

ispositive,theacousticmodeis un-stable and the amplitude of fluctuations grows with time. If

α

is negative, the acoustic mode is stable and perturbations decay withtime. Theeigenvalue procedureisrepeatedby incrementing theamplitudelevel untila limit cycleconditionisreachedwhen

α

=0.

DifferentlyfromSilvaetal. [30]approachforsingleburner se-tups,inamulti-burnerannularcombustorthelinearizationis per-formedforeachoftheFDFsassumedinthemodel.Inthepresent numericalframework,thedistributionofvelocityfluctuationlevels betweenthe FDFs isfixed a prioridepending onthe spinningor standingnatureoftheazimuthalmodeunderinvestigation:

• In a spinning mode, the nodal line rotates at the speed of sound,however,theoscillationamplitudeisuniformallaround the chamber and the velocity fluctuation level

|

u/u

|

is the samefor each burner.In thiscase, the FDFsof the individual burnershavethesamecomplexvalue [25].Mathematicallythis isformulatedasfollows:

|

u/u

|

( )

=C. (6)

Inthis expression,

is a vector containingthe angular coor-dinates ofthe reference pointswhere the velocity fluctuation isspecifiedallowingthecalculation oftheFDF.Eachreference pointliesontheinjectoraxis20mmbelowtheinjectoroutlet. • Inastandingmode,eachinjectoroperateswithadifferent am-plitudeofoscillationdependingonitsrelativepositionwith re-spect to the nodal line [37]. As a consequence, different am-plitudesofvelocityfluctuations

|

u/u

|

havetobeconsideredto representthe response of the flame above each injector. This leads to consider different FDF gains and phase lags values foreach injector.In theproposed methodology, foreach level of velocity fluctuations

|

u/u

|

j, the distribution of

|

u/u

|

over theinjectorsisdeterminedfromthepressuredistribution com-putedbythecodebysetting:

|

u/u

|

( )

=

ψ

( )

/

ψ

max

|

u/u

|

j, (7) where

ψ

(

)/

ψ

max is the normalized azimuthal eigenmode structure.The numerical implementationof thismodelin the

Fig. 1. (a) Photograph of the MICCA combustor with a close up view of a matrix injector and a waveguide outlet. (b) Schematic representation of the experimental setup. (c) Top view of the MICCA chamber with microphone and photomultiplier measurement locations indicated.

Fig. 2. Images of the flame region recorded above one matrix injector under stable operation for the two conditions investigated.

Helmholtzsolverframeworkisnottrivialbecausethemode

ψ

isthesolutionoftheeigenvalueanalysis.Thefrequencyofthis modeandconsequentlyalsothepressuredistributionishighly influencedbytheamountofheatreleaseratefluctuations con-sideredinthemodelasshownin [38](seethetrajectorymaps in Fig.9). Thisresultsin aniterativeprocedure wherethe so-lutionatthekth _{iterationisobtainedusingthe}

_ψ

_distribution computedatthekth_{-1iteration. Thisprocedure isiterated} un-tilthe maximum errordefinedby



=

|

ψ

( )

k−

ψ

( )

k−1

|

is lower than a thresholdvalue, typically 10−3. Itshould be no-ticedthat ateachkth _{iteration,thenonlinear eigenvalue} prob-lemdefinedby Eq.(4)issolved inorderto computethe new modestructure

ψ

[38].

Avalidationofthe proposednumericalprocedure isdiscussed in AppendixA. The caseinvestigatedconsiders an annularcavity withauniformdistributionofheatreleaseandasimplifiedmodel forthenonlinearflameresponsetopressuredisturbances. Analyti-calsolutionswerederivedforbothspinningandstandinglimit cy-clestogetherwithconditionsfortheirstability [16].Confrontations betweennumericalsimulations carriedoutinthiswork and ana-lyticalexpressionsperfectlymatchforboththerotatingand stand-ingmodesvalidatingthenumericalprocedure.

3. Experimentalsetupandnumericalrepresentation

WenowbrieflydescribetheMICCAcombustorshownin Fig.1a. This system comprises an annular plenum connected by 16 in-jectors toan annularchamberformed by twocylindrical concen-tric quartz tubes of 200 mm length. Each injector consists of a cylinder ofdbr=33mm diameterand lbr=14 mm length exhaust-ing gasesthroughal_inj=6mmthick perforatedplatefeaturing89 holesofdp=2mmdiameterlocatedona3mmsquare mesh.The systemfed by a propane/air mixtureallows the stabilizationofa set oflaminar conicalflames above the 16injectors asshownin

Fig. 2forone injector.Twooperatingconditions areinvestigated.

Figure2ashowsflamesstabilizedaboveasinglematrixinjectorfor astoichiometricmixture

φ

=1,whileflamesobtainedforaslightly richer mixture at

φ

=1.11 shown in Fig. 2b have a higher longi-tudinal extension. These images were taken for stable operation

ina singlematrix injector setup. When the MICCA combustor is operated atconditionA, a thermo-acoustic instability coupled to a spinning mode withstable limit cycle is observed [25]. When the MICCA is operated at condition B, a stable limit cycle cou-pledtoastandingmodeisfound [37].Slanted [39] self-sustained combustion oscillations coupled to azimuthal modes were also identified inthissetup when theflow operating conditionswere modified.

The heat release rate is distributed in the simulations over a smallvolume located atthe exit of each burner. It consists of a cylindricalvolume ofheight lf anddiameterdf. Aprevious study onthe same combustorindicates that the dynamicsof perturba-tionsisinfluencedbytheextensionoftheflamedomain [38].For conicalflames,theflamevolumedimensionsarededucedby pro-cessing the images of the flame region under steady conditions shownin Fig.2asdescribedin [38].Thenumericalrepresentation ofthesystemisshownschematicallyin Fig.3a.The plenum con-sistsofan annularcavity linked tothe combustionchamber vol-umebysixteeninjectionunitsasintherealconfiguration.Amodel is used to represent the matrix injectors shown in Fig. 3b. The bodyofeachburner hasthesamedimensionsastherealsystem. The perforated plate is replaced by a cylindrical volume having thesamelinj=6mm thicknessastheperforated plateanda base area witha diameterofdinj=18.9 mmcorresponding to the total flowpassagearea oftheinjector.Thetotalheightoftheburneris l_br+lin j=20 mm. Thecombustion chamber is modeled asan an-nularduct open tothe atmospherewithan augmentedlength of 41mmtoaccountforanendcorrectionresultinginatotallength oflcc=241mm.Thevalueofthiscorrectionwasdetermined exper-imentally by submitting theMICCA chamber to harmonic acous-ticexcitationsneartheunstableresonantmodeandbyscanninga microphonealongalongitudinalaxisabovethecombustion cham-berannulus [38].Allotherboundariesaretreatedasrigidadiabatic walls.Thecombustoroperatesatatmosphericconditions.The tem-peratureoftheplenumissetequalto300K.

Followingpreviousstudies [29,32],thedampingrateisdeduced froma resonance response of the system by imposing an exter-nalperturbationwithaloudspeakerandmeasuringtheresonance sharpness.Thesemeasurementswere carriedoutundercoldflow conditions avoiding any corrections to account for absorption or generationofacousticenergybytheflame [29].This,however, in-troducessomeuncertaintysincethevalueofthedampingrate un-derhotconditionsmaydifferfromthatestimatedatroom temper-ature. Figure 4 shows two acoustic response curves measured in theplenum(Fig.4a)andinthecombustionchamber(Fig.4b) pro-vidingtheresonance frequencybandwidth



fat half-power.The dampingrate

δ

appearingin Eq.(2)isdeducedfromthefrequency bandwidth2

δ

₌



f (s−1).

Theazimuthalthermo-acousticinstabilitiescoupledbyspinning andstandingmodesarediscussedinwhatfollows.Numerical sim-ulations are compared with experiments andthe stability of the numericalpredictionsisevaluated.

Fig. 3. (a) Top view schematic representation of the model. (b) Longitudinal A–A cut showing geometrical details of the matrix injector model and of the flame domain. (c) Three dimensional model of the MICCA chamber with the details of the unstructured mesh comprising approx. 130,0 0 0 tetrahedral elements.

Fig. 4. Acoustic response of the MICCA combustor from 300 Hz to 500 Hz mea- sured by microphones located in the (a) plenum and (b) combustion chamber. The frequency bandwidth f determined at half maximum provides the damping rate in both volumes.

4. AnalysisofoperatingpointA

For operating condition A corresponding to a stoichiometric propane/air mixture

φ

=1 with a bulk flow velocity measured in thecylindrical body ofeach injector equalto ub=0.49 ms−1, the systemsustainsawell-establishedspinninglimit cyclecoupledto thefirstazimuthalmodeatafrequencyof487Hz [25].

Inthenumericalmodel,effectsofsteadycombustionaretaken intoaccountthroughatemperaturedistributioninthegasstream. Thetemperaturewasmeasured,along a longitudinalaxispassing inthecenterofoneburnerlocation,withamovablethermocouple andvariesfrom1470Kneartheflamezoneto1130Kattheend ofthecombustionchamber.Theflamedomainconsistsofa cylin-dricalvolumeofheightlf=4mmanddiameterdf=36mm(Fig.3b). Thesedimensionsare deducedbyprocessingtheimage shownin

Fig.2aasin [38].Thisgivesa flamevolume _Vf=4.18cm3 _which, foraglobalthermalpowerperburnerofQ˙=1.44kW,yieldsaheat releaserateperunitvolumeequaltoq˙ = 3.2× 108_Wm−3_.

Fig. 5. Interpolated Flame Describing Function (FDF) obtained for operating point A: φ= 1.00 and u b = 0.49 m s −1 . Experimental data points are displayed as white dots.

(a) Gain. (b) Phase ϕ.

For each burner, the interaction between combustion and acoustics isexpressedby making useofa globalFDFdetermined experimentallyinasingleburnersetupcomprisingthesame injec-torandequippedwithadriverunit,ahotwireanda photomulti-pliertomeasurevelocityandheatreleaseratefluctuations respec-tively [28,40].Thereferencepointforthevelocityfluctuation mea-surementsislocatedintheinjectionunit20mmbelowcombustor backplane. Figure5showstheFDFusedinthisanalysis. Measure-ments ofthe gain andphase lag were carried out forfive veloc-ityfluctuationlevels rangingfrom

|

u/u

|

=0.1to

|

u/u

|

=0.5(white disksin Fig.5).OnedifficultyistogatherFDFdataatlargeforcing amplitudes.Duetolimitationsoftheequipmentusedtomodulate theflame,itwasnotpossibletocoverthefullfrequencyand am-plituderanges.

A well-resolved FDF is used in the simulations by interpola-tion between the experimental points and extrapolation where experimental samples are missing. At very high amplitude lev-els that are reached in the MICCA experiment, the flames are disrupted and it is reasonable to represent this behavior by a fastdrop intheir response toexternal perturbation. Data are

ex-Fig. 6. Dynamical trajectories in the frequency ( f ) – growth rate ( α) plane colored by the velocity fluctuation level | u  _{/ u | for two damping rates δ= 0 (rectangular}

marks) and δ= 12.5 s −1 (circular marks). (For interpretation of the references to color

in this figure legend, the reader is referred to the web version of this article).

trapolated at these levels with the best fit curve of a smooth fourth order polynomial function using data gathered at lower levels. This is carried out for each forcing frequency. There is a certain amount of uncertainty introduced by this process, but it is based on measured data points while most of the theoretical investigations are based on simplified representations. The non-linear n−

τ

models [41–43], simplified third order polynomials of heat release as a function of pressure [16] or time invariant nonlinearrepresentationsofheatreleaserateasafunctionof pres-sure [24]donotfeatureallthecomplexityofthenonlinearflame responseconsideredinthepresentstudy.Heatreleaserate fluctu-ationsareassumedtobedriven intheMICCAannularcombustor by thefluctuatingmassflow ratesduetoaxial velocity perturba-tions throughthecorresponding injector [3,9].Thisassumptionis reasonableastheinjectorsarewellseparatedintheconfiguration exploredandthereisnovisiblemutualinteraction [27].The refer-encepointsforthevelocity fluctuationsinthenumericaldomain aretakenontheaxisoftheburneratthesamedistancefromthe flamedomainasintheexperiments.

4.1. Dynamicsofaninitiallyspinningmode

In a first stage, following the experimental observations the stability analysis is carried for the first azimuthal mode (1A) of the MICCA chamber starting the simulations with a spinning mode structureandthevelocitydistribution describedby Eq.(6).

Figure 6compares thesystemtrajectoriesplottedina frequency-growth rate plane for two different values of the damping rate

δ

=0s−1and

δ

=12.5s−1 in Eq.(2).Eachcurveiscoloredwith re-specttothevelocity fluctuationlevelthat variesfrom

|

u_/u

|

₌0.10 to

|

u/u

|

=0.63 by steps of 0.01. Symbols are only plotted every 10incrementsandatthetrajectory endpointtoease reading.The dynamical trajectories of the systemare controlled by three free parameters, the frequency f, the growth rate

α

and the relative velocity amplitude level

|

u/u

|

. The 1A mode isfound to be lin-early unstable. For small velocity perturbations, the system fea-tures the highest growth rate of about 280s−1 and a frequency around 400 Hz. A reduction of the growth rate and a substan-tialincreaseoftheinstabilityfrequencyisobservedwhen

|

u/u

|

is augmented.ThisisduetothereductionoftheFDFgainwhenthe velocity fluctuation levelincreases,ascanbe observed in Fig.5a. Ifnodampingisconsidered,thelimitcycleconditionisreachedat avelocityfluctuationlevelthatnullifiesthegainoftheFDF.When dampingisconsideredthelimitcycleisreachedforan amplitude level

|

u/u

|

=0.61,whichcorrespondstoaFDFgainG=0.08ata fre-quency of f=473 Hz. This frequency is close to that observed at limitcycleintheexperimentsf=487Hz.Itisworthnotingthatat the velocity fluctuation level that nullifies the FDFgain, the

sys-Fig. 7. (a) Pressure mode magnitude | ˆ p| with pressure contour lines plotted on a cylindrical surface equidistantly located from the lateral walls. (b) Top: pres- sure structure ˆ p = | ˆ p| cos arg( ˆ p)distribution along the azimuthal direction in the plenum (circular marks) and in the combustion chamber (rectangular marks). Bot- tom: pressure phase evolution in the azimuthal direction in the plenum (circular marks) and in the combustion chamber (rectangular marks).

temshows a negativegrowthrate of

α

=−12.5 s−1 whichequals thedampingratethatwasdeterminedexperimentally.Thischecks thatthedissipationrateiswellrepresentedinthenumerical pro-cedure.

Thestructure oftheunstablemode isnow investigatedatthe limitcycle. Since thesixteenburnershave thesameFDF andare assumedtooperateatthesameamplitudelevel,the circumferen-tialsymmetry of the systemdefined by the annulargeometry is conserved.The nonlinearstabilityanalysisleadstodegenerate so-lutionsfeaturingtwoazimuthalmodessharingthesamefrequency andspatialstructures shiftedby

π

/2asinthelinearcase [44].It isthuspossibletoaddthesetwosolutionsateachamplitudelevel andobtainaspinningmode.

Theresultisshownin Fig.7b(bottom) intheformofa pres-sure phase evolution plotted along the azimuthal direction at a mid-heightposition intheplenumandin thecombustion cham-berbackplane.The phase evolutions featurea shiftofࣃ0.14 rad between the plenum andthe combustion chamber. This angular shiftis alsoobserved inexperiments andconfirmedtheoretically

[25,38].Thepressuremagnitude

|

pˆ

|

shownin Fig.7aisobtainedby plottingpressurecontourlinescomputedbytheHelmholtzsolver onacylindricalsurfacepassingthroughthemiddleofthe combus-tionchamber,theplenum,theburnersandthemicrophone waveg-uides. The pressure iso-linesare deformedin the vicinity of the burnersduetothenearfieldacousticinteractions withthe injec-tors,heatreleasezoneandwaveguides.

However, a spectral analysis of the pressure distribution p= p

(

θ

)

, not shown here, features small harmonic levels. At the burnerslocations, theharmoniccontentremainswithin 6%ofthe signal amplitude and falls to 1% one centimeter away from the chamber backplaneas highlighted by the pressureiso-lines plot-tedin Fig. 7a.This is confirmedobserving the pressure distribu-tionp

(

θ

)

=

|

pˆ

|

cos



arg

(

pˆ

)



alongtheazimuthaldirectionplotted in Fig.7b (top).Deformations appearin thedistribution takenat thebackplaneof thecombustionchamber, whereas the influence vanishesintheplenum.

4.2.Stabilityofthelimitcycle

The stability of the spinning equilibrium point is now inves-tigated followingideasdeveloped in [24]: limit cyclescoupled to spinningmodesarestableifthederivativeoftheFDFwithrespect totheamplitudeofthespinningmodeoscillation(

|

u/u

|

sp)is neg-ative.Foraconstantdampingrate,thederivativeoftheFDFaround theequilibriumpointcanbeapproximatedbythederivativeofthe growthrate,indicatingthat thestabilitycriterioncanbe

reformu-Fig. 8. (a) Four pressure signals recorded by microphones in the plenum (top) and in the combustion chamber (bottom) under spinning limit cycle conditions (dashed lines) compared with numerical reconstructions (solid lines). (b) Spectral content of the pressure signals in the plenum (top) and in the combustion chamber (bottom). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 9. Time resolved heat release rate signals recorded by two photomultipliers in the combustion chamber (dashed lines) compared with numerical reconstruc- tions (solid lines) for | u  _{/ u | = 0.61 (rectangular marks) and for | u}  _{/ u | = 0.58 (circular}

marks). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

latedasfollows:

∂α

∂|

pˆ

|



|u/u|sp <0 (8)

Thegrowthrate

α

plottedin Fig.21casafunctionofthe perturba-tionlevelu_/

|

u¯

|

featuresanegativederivativearoundthespinning oscillationequilibriumpointreachedforu/

|

u¯

|

=0.61. Eq.(8)isthus satisfiedandonecanconcludethatthepredictedspinningmodeis stableasobservedintheexperiments.

4.3.Comparisonswithexperiments

Spinning pressureoscillations atthe limit cyclecorresponding to

|

u/u

|

=0.61arenowcompared tomeasurements.Inthe exper-iments,thesefluctuationsare recordedby microphonesplaced at fourpositionsequidistantlyseparatedontheexternalperimeterof theplenumandatfourpositionsinthebackplaneofthe combus-tion chamber asshown in Fig. 1c. Figure 8(a) (top) displays mi-crophone measurements in the plenum (dashed lines) compared withthenumericalreconstructions(continuouslines)atthesame locations.Thephaseshiftbetweentwomicrophonessignals corre-spondstotheirrelativepositionconfirmingthespinningnatureof themode.Well-establishedsinusoidalsignalswithapeakofabout 260Paatafrequencyof 487Hz,asshownin Fig.8(b)(top),are foundinthe experiments.Theamplitudeandthe phaseshift be-tweenmicrophonesis wellcapturedinthe simulation.Increasing

the examinationperiod, a smallphase mismatchbetween exper-iments andsimulations appears due to the 15 Hzdifference be-tweenthenumericalandexperimentalfrequencies.

Figure 8 (bottom) showsexperimental (dashed lines) and nu-merical signals (solidlines) inthe combustion chamber.The mi-crophonesmountedonwaveguidesatadistanceof170 mmaway fromthebackplaneofthecombustionchambermeasureadelayed signalwithatimelag

τ

m−b=0.5ms.Sincethisdelayisnot negligi-blecomparedtotheoscillationperiodoftheinstability (ࣃ2ms), itistakenintoaccountinthedataprocessing.Itisfirstworth not-ingthatthepressurefluctuationlevelonlyreaches60Panearthe chamber backplaneandexperimentalsignalsare notpurely sym-metricrelativetotheambientmeanpressure,indicatingthe pres-enceofharmonicsasrevealedinthespectralcontentofthe pres-suresignals shownin Fig.8(b) (bottom).One mayhowever note that this harmonic content remains weak. The pressure peak at 974Hzreaches10Pa.Inthenumericalcalculation,signalsare re-constructedbyconsideringonlythefirstharmonicfoundat473Hz inthesimulations.Nevertheless,agoodmatchisfoundintermsof amplitudeandphase forthefoursensors indicatingthat the fun-damentaldominates andprovingthat thenumericalprocedure is abletopredictthedifferenceinamplitudelevelsbetweenthetwo cavities ofthesystem.Again, asmallphase mismatchcan be ob-servedwhenthecomparisoniscarriedoutoveralongerduration. Timeresolvedheat releaseratesignalsare deducedinthe ex-perimentsfromtwo photomultipliers(H1andH2)equippedwith anOH∗filterandplacedatlocationsshownin Fig.1c.Theheat re-leaseratefluctuationQ˙canalsobededucedfromthesimulation:

˙ Q ˙ Q= ˆ u uF

(

ω

r,

|

u/u

|

)

exp

(

i

ω

rt

)

, (9)

where uˆ/u¯ is the calculated velocity oscillation level at limit cy-cle, _{F is the FDF} that needs to be evaluated at the same forcing level,Q˙ isthemeanheatreleaserate,

ω

ristheangularfrequency atlimitcycleandtdenotesthetime. Figure9showstherecorded heat releaseratesignals (dashedlines) andthe numerical recon-structions (solid lines). The two photomultipliers records feature nearly the same amplitudes and a phase shiftof 1.63 rad being close to the theoretical value of

π

/2. However, the amplitude is not constant with time indicating the presence of harmonics. As for the pressure signals, the simulated heat release rate signals onlyconsidertheoscillationatthefundamentalfrequency.The re-constructednumerical signalswith

|

u/u

|

=0.61correspond tothe

Fig. 10. Long time exposure photographs of the flames at a limit cycle coupled to a stable standing azimuthal mode. The velocity nodal line is shown as a dashed red line. (a) V type mode structure with a nodal line observed between burners I –II and IX –X . (b) H type mode structure with with a nodal line between burners VIII –IX and XIV –XV [37] . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Table 1

Sensitivity of the limit cycle oscillation frequency f , the pressure amplitude recorded by microphone MC1 in the chamber, the FDF gain and phase lag and the relative level of heat release rate measured by photomultiplier H1 with respect to the ve- locity fluctuation level | u  _{/ ¯u | reached at limit cycle.}

| u  _{/ u | f (Hz) MC1 (Pa) Gain ϕ( ×π) | ˙ Q} _{/ ˙ Q|}

0.61 473 57.2 0.08 −1.42 0.07

0.60 473 56.3 0.15 −1.42 0.14

0.58 471 54.4 0.22 −1.41 0.26

continuouslineswithrectangularmarksin Fig.9.Theyfeaturethe samephaseshiftasthatrecordedbythephotomultipliers.Interms of amplitude, the two reconstructionssharethe same amplitude, butthevalue isaboutfourtimeslowerthan theonerecordedin theexperiments.

The reasonforthissizabledifference isnowinvestigated with the help of Eq. (9). Giventhe relatively low dampingof the sys-tem (

δ

= 12.5s−1), the limit cycle is reached for a large veloc-ityoscillationlevel

|

u/u

|

=0.61 higherthan0.5, ina rangewhere the FDF gain is slightly extrapolated and features a steep slope as can be seen in Fig. 5a.A small change ofthe velocity fluctu-ation level

|

u/u

|

weakly altersthe corresponding pressure oscil-lation levelinthe plenumandthechamber because these quan-tities scale linearly with

|

u_/u

|

as shown in Table 1. This is not the case forthe corresponding heat releaserate oscillationQ˙/Q˙ duetotherapiddropoftheFDFatlargeperturbationamplitudes. Data in Table1indicatethatareduction of5%ofthelevelof ve-locity fluctuation atthe limit cycleresults ina variation of170% of the FDFgain G and, correspondingly,in the resultingheat re-leaserateoscillationamplitude.ThephaseoftheFDFandthe fre-quencyoftheresonantmoderemainasafirstapproximation unaf-fectedbythesechanges.Theamplitudedifferencesbetween mea-surementsandnumericalpredictionsreducewhentheheatrelease ratesignalsarereconstructedforanoscillationlevel

|

u/u

|

=0.58as shownbythecontinuouslineswithcircularmarksin Fig.9.A per-fect matchinamplitudebetweentheexperimentalandnumerical signals is still not achieved, butdifferences are notably reduced. Thesetestsconfirmthestrongsensitivityofthepredictedlevelof heatreleaseratereachedatlimit cycleduetosmalluncertainties onthevelocityoscillationlevelintheinjector.Thisfeaturereflects thatsmalluncertaintiesonthedatagatheredforFDFathigh forc-ing levels maylead to large deviations of the predictedheat re-leaseratefluctuationsduetotherapiddropoftheFDFgain with theforcingamplitudewhentheflameisdisrupted.

5. AnalysisofoperatingpointB

Foranequivalenceratio

φ

=1.11 andabulk flowvelocityequal to ub=0.66 ms−1, the system features a well-established self-sustained combustion oscillation associated to standing mode at

a frequency of 498 Hz that was fully characterized in [37]. Two standing mode patterns have been observed in the system for the same operating conditions depending on the run recorded.

Figure 10 shows long time exposure photographs of these two

modes with the position of the nodal line indicated by the red dashedline.Themodestructureshownin Fig.10afeaturesanodal linebetweenburnersI–IIandbetweenburnersIX–X.Inthiswork, this mode structure will be designated as “V type”. Figure 10b showsthemodestructurewiththenodallinebetweenburnersV–

VIandburnersXIV–XVthatwillbereferredas“_{H type”.}

The FDF corresponding to this new operating condition is shownin Fig. 11.The samedifficulties to getdataforthe FDFat highamplitudes,typically

|

u/u¯

|

>0.5,persistforthisnew operat-ing condition. As foroperatingcondition A,the heat releaserate is uniformlydistributed over each burner by post-processing the flameimage shownin Fig.2b [38] recordedforoperatingregime Binthesingleburnersetup inathermo-acoustically stablestate. ThisprocedureresultsinaflamevolumeVf=6.11cm3 thatis dis-tributedoveracylinderofheightlf=6mm,2mmlongerthanthat usedforoperatingregimeA,andadiameterofd_f=36mmwhich, foraglobalthermalpowerperburnerofQ˙=2.08kW,yieldsaheat release rate per unit volume equal to q˙=3.31Wm−3. The mean temperatureinthecombustion chambervariesfrom1521 Knear theflamezoneto 1200Kattheendofthecombustionchamber. Thetemperatureoftheplenumissetequalto300K.

Fig. 11. Interpolated Flame Describing Function (FDF) obtained for operating point B: φ= 1.11 and u b = 0.66 m s −1 . Experimental data points are displayed as white dots.

Fig. 12. (a) Pressure modulus | p  _{| of the f = 472 Hz degenerate modes computed un-}

der passive flame conditions and plotted in a plane located at the burner inlet sec- tion. (b) Pressure distribution ψ used to initialize the computation of the standing mode. Symbols indicate the angular position of the sixteen burners.

5.1.Dynamicsofaninitiallystandingmode

Thenumericalproceduredescribedin Section2isusedforthe analysisofthe1Astandingmodedynamicswhen thesystem op-eratesatregime B.Withoutunsteady heatrelease,the circumfer-ential symmetry of the systemdefined by the annular geometry is conserved and the eigenvalue analysis yields degenerate solu-tions. Figure 12a showsthe pressure modulus |p| of the degen-eratesmodes,

ψ

₁(0) and

ψ

₂(0),computedunderpassiveflame con-ditions(q˙₌0) andplottedover a planelocated atthe burner in-letsection.Thetwomodessharethesamefrequency472Hz,but theirstructuresareshiftedby

π

/2.Thecorrespondingdistribution usedtoinitializethesimulationis shownin Fig.12bwhere sym-bolsindicatetheburners’angularlocations.Whenadistributionof FDFsisintroduced(q˙=0),therotationalsymmetry ofthesystem isbroken [16]andtheeigenvalue analysisresultsnomorein de-generateazimuthal modesbutyieldtwodistinctwaves character-izedbydifferentmodalstructures. Thefrequencyandthegrowth rateofthesemodesdependonthelevelofasymmetryconsidered inthesystem.

At theonsetofinstability, Fig.11(b)indicates thatthevelocity fluctuationlevelweaklyinfluencestheflameresponse.The eigen-valueanalysisofthisweaklyasymmetricsystemyieldstwowaves with different yet close frequencies and growth rates (see Ap-pendixB). Increasingthevelocityfluctuationlevel,thedifferences increase between the gain and phase values taken by the FDFs fromthedifferentburners.Thisleadstostrongerasymmetric con-figurationsinwhichthetwo modesresultingfromtheeigenvalue analysisarecharacterizedbyanimportantshiftingrowthrateand alsofrequency.Thefrequencyshiftdependsontheamountofheat releaseratefluctuationconsideredinthesystem [30].Inthe vali-dationcalculationsdiscussedintheAppendix Bwithasimplified heatreleaseresponse,thetwowavesmanifestonlyagrowthrate shift,whiletheysharethesamefrequency.Herethetwosolutions feature both a frequency andgrowth rateshifts. As discussed in

Section 2, foreach velocity level

|

u_/u

|

_j, the mode structure

ψ

_k computedat thekth _{iteration is used in Eq. (7)to modulate the} FDFs.However, inordertobeconsistent,onlythemodestructure closertotheonechosenatthefirstiteration(the

ψ

0

1 or

ψ

20shown in Fig.12)isfolloweduntiltheconvergencecriteriondescribedin

Section2issatisfied.

Thedynamicaltrajectoriesofthesystemarefirsttracked with-outconsideringanydampingbysetting

δ

=0s−1 in Eq.(2)to ana-lyzetheinfluenceofthechoseninitialdistribution.These trajecto-riesareplottedin Fig.13inafrequency-growthrateplaneand col-oredwithrespecttothemaximumamplitudelevel

|

u/u

|

j,which is varied from

|

u/u

|

=0.1 to

|

u/u

|

=0.9. The subscript j will be omittedinwhatfollows.Unlessindicatedotherwise,

|

u/u

|

always referstothemaximumvelocityfluctuationlevelconsideredinthe simulationforthestandingmode analysis.In ordertoemphasize

Fig. 13. Dynamical trajectories colored by the maximum velocity oscillation level

| u  _{/ ¯u | in absence of damping δ= 0 s} −1 . Square marks indicate results obtained for

the H type standing mode structure. Circular marks indicate results obtained for the V type mode structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 14. Dynamical trajectories in the frequency ( f ) – growth rate ( α) plane col- ored by the velocity fluctuation level | u  _{/ u | without damping δ= 0 (circular marks)}

and with damping δ= 12.5 s −1 (square marks) distributed uniformly in the numer-

ical domain. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

thedifferentlevels,symbolsareonlyplottedevery10increments and at the trajectory endpoint. Rectangular marks in Fig. 13 in-dicate results obtained by considering the initial mode structure

ψ

(0)

1 shown in Fig. 12a and designatedasH type.Circular marks in Fig.13indicateresultsobtainedbyconsideringtheinitialmode structure

ψ

₂(0) in Fig.12aanddesignatedasV type.Regardlessof the eigenmode used to initialize the simulations the frequencies andgrowthratesarethesame.Itisworth notingthat,incontrast to spinning mode calculations, a limit cyclecondition cannot be reachedwithout accounting fora finitedamping level regardless ofthe velocityoscillationlevel considered inthesystem. At each point of thetrajectory, flames closeto the nodal linealways ex-perience a small velocity fluctuation

|

u/u

|

. This smalloscillation leadsto a finiteheat release ratefluctuation andthiscauses the modetobecomeunstable.

Withthe introductionof a dampingterm, a limit cycleis ob-tainedasshownin Fig.14wherethedynamicaltrajectories with-out damping,

δ

₌0 s−1 (circularmarks), andwitha dampingrate of

δ

=12.5 s−1 (rectangular marks) are compared. Each curve is colored inthis figureby the velocity fluctuation levelthat varies from

|

u_/u

|

₌0.1 to

|

u_/u

|

₌0.9 withsteps of 0.01. The limit cycle condition

α

=0isreachedinthe simulationfora fluctuationlevel

|

u/u

|

=0.86 ata frequency f=478 Hz. This value is close to that recorded atthelimit cycle inthe experimentsf₌498 Hz.As dis-cussed in [16], the amplitudeofa standing limit cycleis greater than the amplitude of a spinning limit cycle that would settle for the same operating conditions and the same flame response model.Eventhoughtheoperatingconditionsdiffer,itisfoundhere thattheoscillationlevel

|

u/u

|

=0.86ofthelimit cycleof

operat-Fig. 15. (a) Pressure modal distribution | ˆ p| / | ˆ p|max featuring a H type (a) or V type

(b) mode structure calculated by the Helmholtz solver with pressure contour lines plotted on a cylindrical surface passing through the middle of the combustion chamber, the plenum, the burners and the microphone waveguides.

Fig. 16. Velocity distribution at the burners’ positions for the standing mode at limit cycle with the V type (black dotted line) and H type (red dotted line) mode structure. Circular symbols indicate the burners’ positions oriented as shown in Fig. 17 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

ingconditionBassociatedtostandingmodeislargerthantheone

|

u/u

|

=0.61foundfortheoperatingconditionAfeaturingalimit cyclewithaspinningpattern.

In agreement with experiments shown in Fig. 10, two differ-ent modal structures with a nodal line shifted by

π

/2 are pre-dictedatthelimitcycledependingonthechoseninitialcondition.

Figure15ashowsthe_{H type}limitcyclepressuredistribution cal-culatedbytheHelmholtzsolver.Thissolutionisobtainedfromthe modalpressuredistribution

ψ

₁(0)shownin Fig.12asinitial condi-tion. Figure15bshowsthe_{V type}limitcyclepressuredistribution obtainedassumingtheinitialdistribution

ψ

₂(0).Thedistributionof thevelocityfluctuation level

|

u/u

|

reachedatlimitcycleforboth modalstructuresisshownin Fig.16.Itdiffersfromapuresinusoid duetothelocaldeformationsofthepressurefieldnearthe injec-tor outlets as alreadydiscussed for operating conditionA. These deformationsaretakenintoaccountinthe

ψ

functionusedtofix the velocity oscillation level in the differentburners. Symbolsin

Fig.16indicatethelevelsreachedatthesixteenburnerpositions. Onemaynotethatforthelargestvelocityfluctuations,flow rever-sal conditionscan be reachedduringpartof theoscillation cycle intheinjectorslocatedclosetothepressureanti-nodallines.

It is next interesting to compare the location of the pressure nodal line.There is no precise experimental determination of its angularposition.Examiningthelongtimerecordsshownin Fig.10, one finds that inboth configurationsthe nodalline isalways lo-catedbetweentwoburnersinaregionindicatedingreyin Fig.17. The samefigure alsoshowsthe predictednodallineplottedasa reddashedlineforthetwostructures.Forthestandinglimitcycle

Fig. 17. Indication of the predicted angular position of the numerical pressure nodal line (red dashed line) together with the angular region (highlighted in grey) in which the experimental nodal line is recorded for both observed standing mode structures (a) V type (b) H type. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

featuring aV type structure, the nodallineis predictedbetween twoburnersin Fig.17a.Ashiftof

π

/2isobservedfortheH type structurein Fig.17b.

5.2.Stabilityofthelimitcycle

Thestabilityofthestandinglimitcycleisnowinvestigated. Ac-cordingto Ghirardo etal. [24], a standinglimit cycleis stableif threenecessaryandsufficientconditions aresatisfied.Inessence, thefirstconditionrequiresthatatthelimitcycle,thegrowthrate decreases when the velocity level is increased. The growth rate trajectoryobtainedbyvaryingthemaximumamplitudeofthe ve-locityoscillationlevel

|

u/u

|

plottedin Fig.23c(continuouscurve withrectangularmarks)indicatesthatthederivativeofthegrowth ratearound the standing equilibriumpoint is negative. This test confirmsthat the first stabilitycondition is satisfied.The second criterion definesaconditionon theorientation ofthe nodalline. Ghirardo etal. [24] have demonstrated that thiscondition is al-waysfulfilledinconfigurationswithalargenumberofburners. Al-thoughtheMICCAcombustorfeaturesafinitenumberofburners, thissecondcriterionmayberegardedassatisfiedconsideringthat heatreleaseratefluctuationstakeplaceinaflamedomain,which nearly coversthe entiresurfacearea ofthe combustionchamber. The third condition discusses the stability of the standing wave pattern.Stablestandingmodesneedtocomplywiththefollowing inequality: N2n=  2π 0 



F

(

ω

r,

|

u/u

|

i

ψ

(

θ

))

/Z

(

θ

)



sin

(

2n

θ

)

d

θ

>0. (10)

Inthis expression, nis the azimuthal mode order, whichis here equalton=1andthemode

ψ

ischosensuchthatthereisa pres-sureanti-nodeat

θ

=

π

/4.TheH typemodehastheclosest struc-ture fulfillingthis condition and is used for the calculation. The FDFvaluesF in Eq.(10)arecalculatedatthelimitcycleoscillation frequencyf=480Hzwiththecorrespondingvelocitydistribution. ThequantityZ(

θ

)designatestheimpedance,i.e.,theratioof pres-sureintheflamezonetothevelocityatthereferencepoint.

Resultsfor



F

(

|

u/u

|

i

ψ

(

θ

))

/Z

(

θ

)



are plottedin red inFig. 18. Itconsistsofapiecewisefunctiontakingconstantvaluesoverthe angularextensionsoftheflamezonesandzerovaluesinthe angu-larextensionsbetweentwoflames.Inthesamegraphthefunction sin(2n

θ

)isrepresentedinblack.Theconsequenceisthatthe com-ponentinred in Fig.18changessignwithamplituderesultingin apositiveoverallintegralforN2n.Thepredictedstandinglimit cy-cleisthusfound tobe stableasobserved inexperiments.Thisis duetothefact thattheFDFofthematrixburnersinvestigatedin thepresentstudyfeaturesaphaselagwhichissensitiveto

|

u/u

|

Fig. 18. The piecewise function  F(ωr , | u  / u |iψ(θ)) /Z(θ) 

for the combustor un- der analysis (red line) together with a sin (2 n θ) (black line). (For interpretation of the references to color in this figure legend, the reader is referred to the web ver- sion of this article).

asshownin Fig.11b.Thesituationslightlydiffersfromthat inves-tigatedinAppendixAwithasimplifiedflameresponse character-izedbya monotonicallydecreasinggain butaconstant phaselag independentoftheinputamplitudelevel.Inthiscase,limitcycles coupledtostandingwavesarefoundtobeunstable.

5.3.Comparisonswithexperiments

Calculations forthe standinglimitcycleare now comparedto measurements. The frequency and amplitude of velocity fluctua-tions predicted at limit cycle at one referencepoint of the sys-tem are used to reconstruct time resolved pressure fluctuations intheplenumandinthe chamber andheatreleaserate fluctua-tions.Theanalysisisonlycarriedout forresultscorrespondingto the_{V structure} type. Figure 19(a-top) displayspressure measure-mentsintheplenum(dashedlines)comparedwiththenumerical reconstructions(continuouslines)atthesamelocationsinthe nu-mericaldomain.MicrophonesMP3andMP7,locatedat

θ

MP3=78.8° and

θ

MP7=258.7°,respectively,detectpressureoscillationscloseto thenodallineandthey consequently featurelow fluctuation lev-els in the experiments.Microphones MP1 and MP5 located near theanti-nodallinerecordawell-establishedsinusoidalsignalwith apeak amplitudeof350Paatafrequencyof498Hzasshownin

Fig.19(b-top).Giventhestandingnatureoftherecordedmode,the

phaseshiftbetweentwo microphonesignalscorresponds totheir relativepositionwithrespecttothenodalline.Recordsof micro-phoneslocatedatoppositesidesofthenodallineareshiftedby

π

. The amplitudeandthe phase shiftbetweenmicrophones located neartheanti-nodalline,i.e.,MP1andMP5,iswellcapturedbythe simulations.Calculationsalsoretrievetheamplitudesofthe micro-phones located near the nodal line,i.e., MP3 and MP7,however, withasmallphase mismatch.Thisisduetotheshiftof



θ

∼ 3° betweentheangularpositionofthesensorsandofthenodalline shown in Fig. 17a. In the numerical reconstruction, microphones MP3andMP7arelocated onthetwosidesofthe nodallineand their signalsare consequentlyin phaseopposition. Inthe experi-ments, theserecords are nearly in phase asshown by the green and blue dashed lines in Fig. 19(a) (top). Increasing the interro-gationperiodone finds asmall phasemismatchbetween experi-ments and simulations dueto the 20 Hzdifference betweenthe measuredandpredictedlimitcycleoscillationfrequencies.

Figure 19(a-bottom) shows the same type of comparison for pressuresignalsinthe combustionchamber.Again,numerical re-constructions and measurements are, respectively, identified by continuousanddashed lines.Inthe experiments,thesignals fea-tureasome harmoniccontentandsignificantlydifferfromoneto anotherdepending onthe azimuthal positionof thesensors. The analysisofthespectrumcontentofthesignalsshownin Fig.19 (b-bottom)revealsthesecond harmonicpeak at992Hz.A previous studyonthesamecombustorfeaturingstandingmodeoscillations

[37]hasshownthatthisfrequencyisassociatedwiththefirst lon-gitudinal modeofthecombustion chamber(1L).Forthepressure signalslocated close tothe nodal line,theamplitude ofthe sec-ondharmonicpeakiscomparabletotheamplitudeofthe1Amode puttinginevidenceacompetitionbetweenthe0L-1Aplenum os-cillationat495Hzandalongitudinalmode associatedtothe 1L-0Amodeofthechamber.Thisyieldsthedistortedsignalsrecorded by microphones MC3 and MC7 shown in Fig. 19(a-bottom). The other microphones located closerto the pressure anti-nodes fea-ture a second harmonic at 992 Hz with an amplitude of about oneorderofmagnitudelowerthanthatofthefirstharmonic [37]. Thisyieldstheroughlysinusoidalsignalswithapeakamplitudeat 495Hzof60ParecordedbymicrophonesMC1andMC5shownin

Fig.19(a-bottom).IntheFDFframework,thenumericalsignalscan onlybe reconstructedforthefirst harmonicoscillation.Overall, a goodmatchisfoundintermsofbothamplitudeandphaseforthe foursensorsindicatingthatthenumericalprocedureisableto

pre-Fig. 19. (a) Four pressure signals recorded by microphones in the plenum (a) and in the combustion chamber (b) at limit cycle (dashed lines) compared with numerical reconstructions (solid lines). (b) Spectral content of the pressure signals in the plenum (top) and in the combustion chamber (bottom). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 20. Time resolved heat release rate signals for the standing mode recorded by H1 and H2 photomultipliers in the combustion chamber (dashed lines) compared with numerical reconstructions (solid lines).

dictthedifferenceinamplitudelevelsbetweenthetwocavitiesof thesystem. Again, asmallphase mismatchis observedwhenthe comparisoniscarriedoutoveralongerdurationperiod.

The numerical heat releaserate signals (continuous lines) re-constructedwith Eq.(9)arecomparedin Fig.20withthetwoOH∗ light intensitysignals(dashed lines) recordedby the photomulti-pliers(see Fig.1(c)).The twophotomultiplierssignalsfeature dif-ferentamplitudes.Theinjectorclosetothenodallinefeaturesthe largestheatreleaserateoscillationsreachingQ˙_/Q˙ ₌0_.5measured byH1.Adropofabout50%ofthepeakamplitudeisobservedfor theinjectorclosetotheanti-nodalline(H2signal in Fig.20).Itis also worth notingthat thesignal isin thiscasenot apure sinu-soidalwaveindicatingthepresenceofharmonics.Thetwosignals measured by thephotomultipliersare nearly inphase duetothe position ofthe sensors withrespect to the nodalline. Thesmall phase shiftisdue tothe presenceofharmonics. Asforthe pres-suresignals,thesimulatedheatreleaseratesignalsonlyconsider the first harmonic. The reconstructed numerical signalshave the same phase shiftasthose recordedby the photomultipliers.This indicatesthatthephaseoftheFDFiswellcaptured.Thenumerical procedurealsoretrievestheamplitudedropbetweenthetwo sig-nals,however,thepredictedamplitudepeakandtheonerecorded in the experimentsdiffer. For the H1signal, differences between simulations andexperiments aremainly dueto themismatchon

the position of the nodal line shown in Fig. 17. As already dis-cussedforoperating conditionA, theheat releaserateoscillation levelstronglydependsonthepredictedvalue of

|

u/u

|

andonthe gainoftheFDFin Eq.(9).Forhighvelocityfluctuation levels,the FDFgainrapidlydropsasshownin Fig.11a.Asaconsequence,the numerical prediction ofthe H2 signal is quite sensitive to small uncertainties on the velocity oscillation level at the limit cycle. These uncertaintieslead to important variationsof the FDF gain Gand consequently in thepredicted heat releaserate oscillation atlimit cycle. The origin ofthe differencesobserved at limit cy-clesbetweenmeasurements and simulations isthus the sameas forthespinningmodecalculationsstudiedintheprevioussection.

6. Modetypeselection

Followingthesuggestionofa reviewer,aninvestigation ofthe possiblescenariosleadingtothespinningandstandinglimitcycles analyzedintheprevioussectionsisnowproposed.

6.1. AnalysisfortheoperatingpointA

The stability analysis is repeated for operating condition A, but instead of assuming an initially spinning mode as done in

Section4.1,simulations areinitiatedwithastandingmode struc-tureandthevelocity distributiondescribedby Eq.(7).The corre-spondingtrajectory(blackcontinuouslinewithrectangularmarks) plottedinafrequency-growthrate-velocityfluctuationlevelspace iscompared in Fig.21awiththe spinningmode trajectory calcu-latedin Section 4.1(redcontinuous linewithcircularmarks). At thestartingpoint,i.e.for

|

u_/u

|

₌0.1, thetwotrajectoriesperfectly match. This is due to the fact that in the linear regime, i.e. for valuesof

|

u/u

|

≤0.1, theFDFissimplyatransfer functionandits gainandphase donotdepend ontheinput amplitudelevel.This alsomeansthatinsimulationsstartedwithastandingmode struc-ture, each burner ofthe chamber operates with thesame veloc-ityfluctuation.Projectingthesetrajectoriesonafrequency-growth rate(f −

α

)plane,asshownin Fig.21b,onemaynotethatwhen

|

u/u

|

isprogressivelyincreasedthe twotrajectoriesremain close andalimitcycleconditionisreachedforbothspinningand stand-ing modes for a frequency value ofabout 473 Hz. In this graph each lineiscolored withrespect tothe value assumedby

|

u/u

|

. Differences in trajectories are, however, made visible in Fig. 21c

Fig. 21. (a) Trajectories f −α−| u  _{/ u | of solutions of Eq. (2) for operating regime A initiated with a standing mode structure (black continuous line with rectangular marks)}

and a spinning mode structure (red continuous line with circular marks). Projection of the trajectories on f −αplane (b) and α−| u  _{/ u | plane (c). Circle markers indicate}

results for an initially spinning mode structure, square markers indicate results for a standing mode structure. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).