Azimuthal flame response and symmetry breaking in a forced annular combustor

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Combustion and Flame

journalhomepage:www.elsevier.com/locate/combustflame

Azimuthal flame response and symmetry breaking in a forced annular combustor

Håkon T. Nygård

^a,∗

, Giulio Ghirardo

^b

, Nicholas A. Worth

^a

aDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim N-7491, Norway

bDoodle, Werdstrasse 21, Zürich 8021, Switzerland

a rt i c l e i nf o

Article history:

Received 19 November 2020 Revised 11 June 2021 Accepted 11 June 2021

Keywords:

Azimuthal modes Combustion instabilities Acoustic forcing Bloch theory

Azimuthal flame describing functions

a b s t r a c t

Inthecurrentstudyazimuthalforcingofanannularcombustorwithswirlingflameshasbeenperformed topresentforthe firsttimetheHeat ReleaseRate(HRR) responsetoallpossiblepressure fieldsofthe firstazimuthalmodeuptoafiniteamplitudelimit.Theresponseisfirstquantifiedthroughtheconven- tionalFlameDescribingFunction (FDF)framework,showingadifferenceinresponsewhichdependson whethertheacousticfieldrotatesanti-clockwiseorclockwise,albeitwithsomescatter.Additionallyand somewhatsurprisingly,afiniteHRRresponseisobservedforflamesexactlyinthepressurenode.AnAz- imuthalFDFisintroduced,basedonthedecompositionofthespatialHRRresponsethroughBlochtheory, tobetterhighlightthedifferenceinHRRresponsetotheanti-clockwiseandclockwisecomponentsofthe acousticfieldandreducescatter.Acleardifferenceinresponseisobserved,withasignificantlyhigher responsetotheanti-clockwiseforcingcomponentcomparedtotheclockwisecomponent,independent oftheprescribedpressuremode.Thedifferenceisattributedtothesystematicsymmetrybreakingintro- ducedbyusinganannularenclosureoffinitecurvatureandwidthwithswirlingflames.Itisarguedthat thefinitecurvatureandwidth ofthegeometryand theswirlneedtobebothpresenttoobservethis effect.ThedifferenceinresponseresultsinadifferencebetweenthenatureangleoftheHRRmodeand thatoftheacousticfield,explainingtherelativelylargeHRRresponseobservedforflamesinthepressure node.TheAzimuthalFDFdescribeallofthesephenomenawell,andisthereforeconsideredmoresuitable thantheconventionalFDFtocharacterisetheresponseinanannularcombustor.

© 2021TheAuthor(s).PublishedbyElsevierInc.onbehalfofTheCombustionInstitute.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Thermoacousticinstabilitiesare awell knowndevelopmentis- sueforgasturbineengines[1].Theseoccurduetotheinteraction betweenheatreleaserateandpressureoscillations,andarepreva- lentwhenoperatingoververywide rangesofconditionsorwhen varying fuels;bothofwhichmaybeadvantageous forthecontrol ofemissionsinmodernengines.Thus,inordertorealisethebene- fitsofincreasedfuelandoperationalflexibility,accurateprediction ofthermoacousticstabilityisimportant.

One promising approach to predict the thermoacousticstabil- ityofasystemistouseeitherlow-orderacousticnetworkmodels [2–6] orHelmholtzsolvers [7–9].Theseemploy FlameTransferor Flame DescribingFunctions(FTF/FDF)inordertocouplethenon- linearflameresponsetotheacousticsofthesystem,andareusu- allyobtainedexperimentally [10,11],ornumerically[12,13].While

∗Corresponding author.

E-mail address: [email protected] (H.T. Nygård).

thismodellingapproachhasbeenappliedsuccessfully,theuseofa responsefunctiontodescribetheflames gainandphaseresponse meansthattheaccuracyofanystabilitypredictionsisdirectlyde- pendentontheapplicabilityofthisfunctiontothesystem.

Practicalgasturbinecombustorsoftenfeaturemultipleswirling flamesarrangedaroundanannularchamber.Theadventoflabora- toryscalecombustors [14–16]hasmadeit possibleto study self- excitedthermoacousticinstabilitiesinfullannulargeometries[17]. Themostprevalentmodesofexcitationinthesechambersareaz- imuthal in nature,which can induce acoustic oscillations in both azimuthal and axial directions. The lack of azimuthal acoustic boundaries results in degenerate modes, which can be charac- terised conveniently through the recently introduced hypercom- plex quaternion formalism [18]. This describes the acoustic field in terms of four state space variables: the mode amplitude, A; the orientation angle of the anti-nodal line,

θ

0; the nature an- gle,

χ

^{,whichdescribesifthemodeisspinning,standingorsome}

combination thereof; and the temporal phase

ϕ

^{. Previous stud-}

ies have shown that for a given operating condition the self- excitedpressuremodecanexhibitpreferredcombinationsofstate

https://doi.org/10.1016/j.combustflame.2021.111565

0010-2180/© 2021 The Author(s). Published by Elsevier Inc. on behalf of The Combustion Institute. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Nomenclature

A Amplitude of azimuthal pressure fluctua- tions[R,Pa]Eq.(5)

A_st Amplitude ofthe standing componentof the azimuthal pressure fluctuations [R, Pa]Eq.(6)

f Oscillationfrequency[R,Hz]Fig.3 FDF^±

ˆ u_axial

±

Azimuthal Flame Describing Function of ACW (-) and CW (+) component [C,-]

Eq.(20) FDF_j

ω

,

^uˆaxial,j

Conventional Flame Describing Function ofthe jthflame[C,-]Eq.(7)

N Numberofinjectors[N⁺,−]Fig.1 p(

θ

,t) ^{Azimuthal pressure} fluctuations [R, Pa]

Eq.(5)

p_d(^x,t) ^Pressure fluctuationsin the injector tube [R,Pa]Eq.(1)

q(^r,

θ

,t) ^{HRR on a pixel by pixel basis [R,} W]Eq.(8)

q(^r,

θ

,t) ^{Phase dependent} fluctuationsin HRR [R, W]Eq.(8)

^q¯(^r,

θ

)

sectors Rotation average of temporal mean HRR.

Reduces the spatial r and

θ

^dependence

from the full annulus to a single flame sector.[R,W]Eq.(20)

qˆ

j Fourier amplitude at the peak frequency of the spatially sector integrated HRR of the jthflame[C,W]Eq.(7)

T Oscillationperiod[R,s]Fig.3

t Time[R,s]Eq.(1)

t₀ Chosenstart timeofphaseaverage[R,s]

Eq.(9)

t_start Arbitrarystarttime[R,s]Fig.3

U_bulk Bulkvelocityininjectortubeevaluatedat thedumpplane[R,m/s]Eq.(7)

u_axial,j(^t) ^{Axial acoustic velocity} fluctuations (u_d) evaluatedatthedumpplaneatazimuthal injectorlocation

θ

j[R,m/s]Eq.(7)

u_axial(^t)

±

j Shorthand for the rotation average axial acoustic velocity at the dump plane for the jthinjector[R,m/s]Eq.(19)

u_axial^,rec_,j(^t) ^{Shorthand for the rotation average re-} constructed axial acoustic velocity atthe dump plane for the jthinjector [R,m/s]

Eq.(19) Greeksymbols

θ

^{Azimuthal angle in the combustion chamber [R,}

rad]Eq.(5)

θ

0 Orientationangleoftheazimuthalpressurefluctu- ations[R,rad]Eq.(5)

θ

0,q Orientationangleofsector integratedHRRfluctu- ations[R,rad]Fig.15

θ

j Azimuthal angle of the center of injector corre- spondingtothe jthflame[R,rad]Eq.(7)

χ

^{Natureangleof thepressure}fluctuations[R,rad]

Eq.(5)

χ

^{q NatureangleofsectorintegratedHRR}fluctuations [R,rad]Eq.(25)

ψ

±1(^r,

θ

) ^{Bloch kernelsof thefirst azimuthal mode [}C, W]

Eq.(12)

Modifiers

|

(·)

|

^{Absolutevalueof}(·)^[R]Eq.(7)

(·)^rec Reconstructionof(·) ^{fromrotationaveragecom-} ponents.Theresultingvaluesregainthefullspa- tialrand

θ

dependence.Eq.(16)

(·)a Analyticalsignalof(·)^[C]Eq.(8)

(ˆ·) ^{Fourier amplitude of} (·) ^{at peak frequency. Re-} movesthetimedependence.[C]Eq.(7)

(·)

annulus Spatialaverageof (·)^{overthe annulus.Removes} allspatialdependence.Fig.10

(·)

j Spatial average of (·) ^{over the jth flame sector.}

Removesspatialdependencewithinthe jthflame sector.Eq.(7)

(·)

^{± Rotationaverage inthe ACW (-)andCW(+) di-}

rectionof(·)^{.Reducesthespatialrand}

θ

^depen-

dencefromthefullannulustoasingleflamesec- tor.Eq.(14)

(·)

sectors Rotation average ofthe temporal meanquantity (·)^{.Reducesthespatialrand}

θ

^{dependencefrom}

thefullannulustoasingleflamesector.Eq.(11) Abbreviations

ACW Anti-clockwise CW Clockwise

FDF FlameDescribingFunction HRR HeatReleaseRate

LPHR LowPerturbationHighResponse

space parameters A,

θ

0 and

χ

^{[17,19], and moreover, that these}

can also vary slowly with time, resulting in distinct modal dy- namics[14,15,20,21].Furthermore,thepresenceofmultipleflames around an annular chamber means that neighbouring flames are freetointeractwitheachother[22],addingfurthercomplexityto theresponse.

Atpresent,FDFsarecommonlymeasured,simulatedordefined basedon isolated singleflamesetupssubjected to acousticoscil- lationsin theaxial,orbulk flow, direction[10–13,23].Such func- tions are therefore suitable for predicting the stability of single sectorrigsinordertodemonstratethemethodology[24–26],and may also be suitable for describing the stability of more prac- tical can-annularconfigurations [27]. However, transfer functions obtainedfromsinglesectormeasurementsomitby definitionfea- turesassociatedwiththe morecomplexresponsefound inannu- larconfigurations,andincludenofunctionaldependenceon

χ

^or

θ

0,whichisapotentialsourceofuncertaintyinsuchstabilitypre- dictions. While a handful of studies have used either simplified transferfunctionmodels,orfunctionsobtainedfromsinglesector measurements topredict the stabilityfeatures ofan annularsys- tem[9,28–34],theuseofsingleflametransferfunctionsmeansany possibleeffectsduetoadjacentflamesinteractingwitheachother are eliminated. Additionally, any effects on the transfer function fromthebulk swirl,duetothe geometry,combinedwiththeaz- imuthalmodearenotincluded.Itthereforeremainsanopenques- tionwhethertransferfunctionsobtainedinsingleflamesetupsare more generallyapplicable forstability predictionin annularcon- figurations,asthefunctionaldependenceoftheseon

χ

^or

θ

0isas yetunknown.

Inordertobetterunderstand theflameresponsetoazimuthal oscillations,anumberofstudieshaveexaminedsingleflamescon- fined within enclosures which are long in the transverse direc- tion(orthogonaltothebulkflow)[35–40].Sucharrangementsare useful as they allow the effect of transverse acoustic excitation tobe studiedacross awide rangeoffrequencies,whileremoving thecomplexityassociatedwithflame-flameinteractions.Morere- cently,transverseoscillationshavealsobeeninvestigatedinlinear

Fig. 1. The externals of the annular combustor is shown in (a) with the forcing array mounted above the flames to reduce the direct effect of the standoff tubes on the flames. The schematic of the dump plane as viewed from the downstream direction in (b) shows the arrangement of the N = 12 injectors. Each instrumented injector is indicated by an arrow and denoted P j. The swirl direction is indicated by the red arrow, and one flame sector is indicated by the grey shaded area, azimuthally centered at the injector center line. The side view of an instrumented injector, shown in (c) , indicate the pressure transducer and swirler locations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

arraysofmultipleflames[41],permittingtheeffectofflame-flame interactionstobestudiedinadditiontotheeffectoftransverseos- cillationsontheflamedynamics.However,allsystemswithlinear enclosuregeometryomiteffectsassociatedwithfinitewallcurva- turecombined withabulk swirlintheannulus inducedby indi- vidual swirlingflames,prohibitingtheinvestigationofthesesym- metrybreakingeffects.

Anumberofrecenttheoreticalstudieshaveexploredsymmetry breakingeffects,throughthederivationofdynamicalequationsfor the azimuthal modestate spacevariables[42,43].Thedegenerate eigenvaluehasbeenshowntosplitwhenthereflectionalsymme- try is broken [44], leadingto two differentgrowth rates [43,45], but these effects are not related to changes in the heat release rate. While the describing function in these dynamical formula- tionsisallowedtohaveadependenceonthenatureangle

χ

^,this

hasnotbeenwidelyexplored,asthereisasyetnosystematicev- idence linking the nature angleand heat release rate. Therefore, whileitiswidelyknownthatazimuthalmodescancontainarich rangeofmodelresponses,andthereexistsawaytoincludethese indynamical equationsgoverningthesystemstability,there isas yetlittleguidanceonhowtolinkthedependenceoftheresponse functiontotheazimuthalmodenature.

In order to address this the current study is aimed at cap- turing for the first time the full dependence of flame describ- ing functions in an annular geometry on the nature, orienta- tion, and amplitude of azimuthal modes. In order to exert con- trol over these parameters, the excitation strategy of [46] is ap- pliedinanannularcombustor.Thisapproachhasbeenpreviously demonstrated for the excitation of standing[19] and single am- plitude spinning modes [47], but the full range of spinning and mixed standingandspinningmodes havenot been explored yet.

The present study realises a 40 fold increase of the number of parameter combinations compared to previous work, which al- lows a detailedexamination ofthe functionaldependence ofthe FDF on nature angle, orientation angle andamplitude leadingto newandsomewhat unexpectedresults.Differentspatiallocations within a transversestandingmode [48,49] anda transversetrav- elling acoustic mode [38] have been studied in single flame se- tups, and it has been shown that small asymmetries in single flamescancauseadifferenceinresponsewhensubjectedtotrans- verse acoustic velocities [50–52]. However, this study isthe first to examine theeffectof thesystematicsymmetry breaking asso- ciated with the presence of swirling flames in an annular com- bustion chamber. The analysis is performed ina waythat mini-

mizes theeffectsofsmall flametoflame differences,shifting the focusfromtheeffectonasingleflametotheeffectontheglobal response.

Itisalsoworthnotingthatthesemeasurementsalsopermitex- amination offlame to flamedifferences, by comparing nominally identicalforcedstateson12nominallyidenticalflamespositioned around the annular chamber. Finite physicaldifferences between individualflamesareanothersourceofsymmetrybreaking,which hasbeen showntoaffect systemstability[53].Toanalyse this, a novelimplementationoftheBlochformalism[8,54]isintroduced, enablingthecalculationofanaverageflame(whichisindependent offlameto flamedifferences), andthedecompositionofthe first azimuthalmode intotheindividualACW andCWcomponents.In this manner the FDF of a single, average flame can be defined.

Additionally the decomposition is used to define a so calledAz- imuthalFlameDescribingFunction(FDF^±)foreachcomponentspin- ningACW(-)andCW(+)ofthemode,whichisshowntobethe mostdescriptive representation ofthe heat releaserate response inthecurrentannularconfiguration.TheAzimuthalFDFs arealso used to provide a theoretical, nature angle dependent HRR de- scription,suitable foruseintherecentlyderiveddynamical equa- tions[42,43].

The rest of the paper is organised as follows. First the ex- perimental setup and post processing is described in detail in Section 2, startingwiththe setup (Section 2.1), pressure calcula- tion methods (Section 2.2), conventional Flame Describing Func- tiondefinition(Section2.3),howtoprescribethepressuremodes (Section 2.4), phaseaveraging(Section 2.5),beforefinallydescrib- ing rotational averaging and the corresponding Bloch waves for- malism(Section2.6)andintroducingtheconceptoftheHRRmode (Section2.7).ThemeasuredFDFsoftheindividualflamesarepre- sentedin various forms inSections 3.1–3.3, followed by the two Bloch kernels in Section 3.4. The new Azimuthal FDFs are pre- sentedin Section 3.5, beforethe implications onthe HRRmodes arediscussedandquantifiedinSection3.6.Lastlytheconclusions aregiveninSection4.

2. Experimentalmethods 2.1. Geometryanddataacquisition

The combustor used in this study is the annular combustion chamberusedin[47]intheN=12flameconfiguration,asshown in Fig. 1.The combustor is described in detail inearlier studies,

butthe mainfeatures anddimensionswillbe repeatedhere. The combustion chamber consists of two concentric cylindrical walls ofdiameterdouter=212mmandd_inner=129mm.Tobecomparable totheself-excitedconfiguration,andhelpwithopticalaccessfrom above, theinnerwall isshorterthan theouter wall,withlengths of127mmand287mmrespectively.Bothwallsaremountedatthe dump plane,whichisdefinedastheplane separatingtheinjector tubesandthecombustionchamberasshowninFig.1c.Aspeaker array, consistingof 8 equidistantly spaced,68mm long standoffs, is mounted with the speakers approximately 104mm above the dump plane, ensuring the speakers do not interact directly with theflamesthroughvelocityoscillationsinducedattheexitsofthe standoff tubes. Four of the standoffs are equipped with Adastra HD60horndrivers,whichareusedtoprescribetheforcedstateon the system. Theremaining standoffs areleft blockedat theouter endofthearray,marginallyimprovingthecontroloftheresulting forcedstate.The horndriversare drivenbyan Aim-TTITGA1244 signal generator,amplifiedby apairofQTXPRO 1000poweram- plifiers.

Eachofthe12equidistantlyspacedinjectorshaveadiameterof 19mm,withacentralbluff bodyofdiameter13mmandhalfangle 45^◦.Thebluff bodyismountedtoarodofdiameter5mmusedfor centeringandactsasamountingpointforaswirler.Theswirleris mounted10mmbelowthedumpplane,measuredfromthetrailing edgeoftheswirlervanes,andinducesanACWswirlwhenviewed fromdownstream.Theresultingswirlnumber10mmdownstream of the dump plane has beenmeasured to be 0.65 foran uncon- finedconfiguration.Theinjectorsare150mmlongandtheyarefed byaplenumwithflowstraightenersandahemisphericalbodyfor equalflowdistribution.Theplenumisfedbytwoimpingingjetsof premixedairandethylene,atanequivalenceratioof=0.7.The flow rateandequivalenceratioiscontrolled bythree Alicatmass flow controllers,andtheflowrateissettogiveabulk velocityof U_bulk=18m/satthedumpplane.Thisresultsina thermoacousti- cally stableoperatingconditionforthestudiedconfiguration,eas- ingthecontroloftheforcedazimuthalmode.

Six of theinjector tubes are instrumented with two differen- tial pressure transducers of the type Kulite XCS-093-0.35D flush mountedwiththeinjectorwall,spacedby65mmintheflowdirec- tion.ThesignalsareamplifiedbyapairofFyldeFE-579-TAbridge amplifiers anddigitized by a set ofNI-9234 DAQ modules, oper- atingat51.2kHz.Heatreleaseratedataisobtainedby aPhantom V2012highspeedcameraequippedwithaLaVisionIntensifiedRe- lay Optics(IRO) unit. The IRO is equippedwitha Cerco2178 UV lens with a narrow band pass filter, centered at 310nm with a full width halfmaximum of10nm.This capturesthe light inten- sitywitha wavelengthcorresponding tode-excitingOH^∗radicals, whichhasbeenshowntobeproportionaltotheheatreleaserate for perfectly premixedcombustion [55].The imaging system op- erates at10kHz,whichis sufficienttimeresolution forthe forced statesofinterest,andtheIROgatetime is80

μ

^{s.The samplerate}

ofthesystemisnotamultipleofthefrequencyoftheforcedstate by design, improvingthe numberofphase instances capturedby the camera. The lengthened trigger signal of the IRO unit is ac- quired onthe samesystem asthepressure transducers, enabling synchronization of the pressure and heat release rate signals. A spatial resolution of 2.5 pixels per millimeter is achieved and a minimumof20,000imagesaretakenforeachforcedstate.Atotal of5,250,000forcedflameimagesweretakenaspartofthecurrent study.

2.2. Modereconstruction

Duetothelongaspectratiooftheinjectortubes,andsmalldi- ameterrelativetothewavelengthoftheacousticmode,theacous- tic pressurefluctuationsin thetube aretwo counter propagating

Fig. 2. Frequency spectrum measured by the upper microphone in the injector at θ= 0 ^◦at different am plitudes of ACW forcing, ensuring similar magnitude pressure oscillations at all azimuthal locations. Amplitudes range from 1224 Pa for the High forcing level, through 661 Pa ( Middle ) and 340 Pa ( Low ) to the noise level ( < 10 Pa ) for the thermoacoustically stable Unforced case. Due to the sharp peak at the prescribed forcing frequency, the frequency of oscillation will be approximated to be constant.

planewaves, p_d

(

^x,t

)

⁼

[B₊e^−ik+x+B₋e^ik−x]e^iωt

, (1)

where denotes the real part of its argument

{

...

}

^{. The ampli-}

tudeofthetwocounter propagatingcomponentsaregivenbyB₊ andB₋,correspondingtothecomponentpropagatinginthedown- streamandupstreamdirectionrespectively.xis theverticalposi- tioninthetubeasshowninFig.1.Thewavenumbersk_±aregiven by[56]

k_±=

ω

^/c

1±U/c= k0

1±Ma, (2)

where c is the speed of sound in the medium and Ma is the Mach number. The left hand side of Eq.(1) is measured at two discrete locations, xupper=−44mm and x_lower=−109mm, by the pressuretransducersinthe instrumentedinjectors.The analytical signal of p_d(^t)^{is then obtainedthrough theHilberttransform} H of the measured signals, p_d,a(^xl,t)=p_d(^xl,t)+iH p_d

(^xl,t)^{. The} acoustic velocity perturbations corresponding to the pressure in Eq.(1)can be derived fromthefluctuating momentum equation, andaregivenby[56]

u_d

(

x,t

)

= 1

ρ

^{c B+e−ik+x−B−eik−x}

e^iωt

, (3)

The expression forthe velocity in Eq. (3) is calculated from the pressuremeasurements atthetwo microphonelocationsby solv- ingEq.(1)forB_±e^iωt.

Ingeneral,thesolutionofEqs.(1)and(3)isperformedforall frequencies, but asa computational simplificationthe prescribed forcing frequencyis assumedto bethe only frequencyofsignifi- cance.Thisassumptionisbasedontheamplitudespectrumshown forafewamplitudesoftheforcedstatesandanunforced casein Fig.2.Theamplitudeattheforcingfrequencyof f=1650Hzissev- eralordersofmagnitudeabovethebackgroundnoiselevel,andthe firstharmonicisabouttwoordersofmagnitudelowerforthepre- sentedcases.Combinedwiththeuseofthesameequivalenceratio andinletvelocity forallforcedstates,theassumption ofasingle frequencyisreasonable. Fora constant crosssection,propagating the pressure and acoustic velocity fluctuations to alternate loca- tions is performed by inserting the location x into Eqs. (1) and (3). In the case of smooth area changes, A1→A2, the pressure andmassare conserved acrossthe jump inthezero Machnum- berlimit[2]

p_d

2

1=0 and Au_d

2

1=0. (4)

Theswirlerwillintroduceaninitialarea decrease,followedbyan increasebacktotheoriginalarea.Thistemporaryareachange,and

Fig. 3. Example of reconstructed pressure signals at the dump plane at three fixed azimuthal locations for an ACW spinning mode ( top ) and a standing mode ( bottom ) over three oscillation periods. The separate starting phases of the two cases are arbitrary. The curves are calculated from Eq. (5) with parameters determined from two different forced states.

other potential effects ofthe swirleron the axial acousticmode, areincludedthroughtheuseoftheexperimentallymeasuredscat- teringmatrix[57,58].Theareachangeintroducedbythebluff body isestimatedas10separatestepchangesinthediameter,eachwith the samelength and volumeasthe corresponding section ofthe injectorandbluff bodysystem.

After propagating thepressure oscillations to the dump plane of the combustor, theinjector tubes terminate inan annularge- ometry with an azimuthal pressure mode. The acoustic mode in theinjectortubescouplesprimarilytotheacousticpressureinthe chamber andnottotheacousticazimuthal velocity[59,60].Since all theinjectorsare nominallyidentical,andthe potentialdepen- dence of the impedance on the mode is low [59], it is safe to assume the impedance linking the pressure in the injectors and dump plane is thesamefor allthe injectors.Neglectingthe spe- cificvalueofthepressurecouplingimpedance,thepressureinthe combustion chamber can be describedby the following pressure ansatzdescribingthefirstazimuthalmode[18]

p

( θ

^,t

)

=Acos

( θ

⁻

θ

⁰

)

^cos

( χ )

^cos

( ω

^t+

ϕ )

+Asin

( θ

⁻

θ

0

)

^sin

( χ )

^sin

( ω

^t+

ϕ )

^{. (5)}

The derivation based on quaternions and detailed procedure for fitting the measured pressure at the dump plane can be found in[18].Inthecurrentstudyaleastsquaressolutionbasedonthe sixpropagated pressuresare usedtoreduce theeffectofrandom fluctuationsinthepressuresignal.Theamplitudeoftheazimuthal pressuremodeisdenotedAinEq.(5),and

χ

^{isthenatureangleof}

themode.Thenatureangledescribeswhetherthemodeispurely spinning (

χ

=

π

/4 for ACW,

χ

=−

π

/4 for CW), purely standing (

χ

=0),oramixofthetwo forintermediate values.It isequiva- lenttotheSpinRatio(SR)[15]usedinpreviousstudies[14,47]on the same geometry through the relation SR=tan(

χ

)^{. The orien-} tation angle

θ

0 gives thelocation ofthe pressure anti-node, and

ϕ

^{isatemporalphase.Thepressure}fluctuationsinEq.(5)canbe considered as a combination of a spinning and a standing wave component, wheretheorientation angle

θ

0 describestheorienta- tionoftheanti-nodeofthestandingcomponent.Theamplitudeof thestandingcomponentisgivenby

Ast=A

1−2

|

^sin

χ |

^cos

χ

. (6)

Thecharacteristicfeaturesofthepressuremodeforaspinning andastandingmodeare shownfortwo exampleforcedstatesin Fig. 3. The spinning mode in the top row show all the pressure locations havesimilar amplitude, andthephase difference corre- spondstothetime ittakesthewave tocoverthespatialdistance between the pressure transducers. The standing mode is in con-

trast observed to have two similar amplitudepressure signals of oppositephase,withathirdsignalofnegligibleamplitude.Thisis causedbythecharacteristicalternatingpatternofnodesandanti- nodes,equidistantlydistributedeveryazimuthalangle

π

/2forthe first azimuthal mode. For the range ofnature angles

χ

^between

thestandingandthespinningmodes,themodecanbeconsidered amixedmode,withastandingcomponentandaspinningcompo- nent.Theconceptoftheorientationangle

θ

0 onlymakessensein thecaseofmixedorstandingmodeswhereAst=0.

2.3. Flamedescribingfunctions(FDFs)

The interaction between a single flame and acoustics are of- ten described by the conventional Flame Describing Function (FDF) [61] of a single flame, where the spatially integrated HRR ofthe flameis considered.In the currentconfigurationthere are N=12differentflamesinsidethecombustionchamber.Toextract theresponsemostequivalenttoasingleflamesetup,theresponse within oneflame sector, asdefinedinFig.1 b,isconsidered. The spatiallyintegratedHRRofeachflamesector,denoted

^q

jforsec- tornumber j,isobtainedbysummingover thepixelswithin the sector. Thisremovesthespatialdependencewithin the jthflame sector,quantifying theHRRofasingleflameasascalar.Thecon- ventionalFDFforthe jthindividualflameintheannularcombus- torfromtheinjectorlocatedat

θ

=

θ

jisthendefinedas[61]

FDF_j

ω

^,

uˆ_axial,j

=

qˆ

j/

^q¯

j

uˆ_axial,j/Ubulk

, (7)

whereuˆ_axial,j istheaxial velocityperturbationfromEq.(3)inthe jth injector at a reference axial location, here chosen to be the dump plane.

qˆ

j is the complex Fourier amplitude of the heat release rate oscillations at the prescribed angular excitation fre- quency

ω

=2

π

^{f,andthemeanheatreleaserateofthe jthflame}

isdenoted

^q¯

j.Inrecentwork[43,53,62]similarapproacheshave been used to describe the interaction between heat release rate andpressureinannularconfigurationsinthegoverningequations.

Usuallythedescribingfunctionisobtainedoverarangeoffre- quencies f forlongitudinal forcingsetups. However, inthisstudy onlytheresponseofthefirstazimuthalmodeisofinterest,mean- ingthe frequencywillbefixed bythe geometryandtemperature ofthecombustionchamber.Theresponseisstill definedasa de- scribingfunction[63,Section2.3],andthenamingisinlinewith the first useof describing functionsin thermoacousticinstability studiesbyDowling[3,Eq.(3.8)].Thestudyofasinglefrequencyis incontrastto one previousstudyon asimilar configuration[64], wherea large range offrequencies were used, andthenature of the excited modes was not explicitly controlled, making the re- sultsmore difficultto interpret.Instead inthe presentstudy,the describing function ata fixed frequencywill be explored fordif- ferent combinations of the state space variables A,

χ

^and

θ

0 in Eq.(5)forsixdifferentflamesdirectlyfromexperimentalmeasure- ments,andfrom12 flamesby interpolation ofthepressurewave oscillations.Thisisthefirsttime thedescribingfunctionisinves- tigated for12 nominally identicalswirling flames arranged inan annulus,enablingassessmentofthevariabilitybetweentheinjec- tors.Therefore, theflame toflame differencescanbe assessed in detail.

2.4. Prescribingforcedstates

Previous experimental studies featuring self-excitedazimuthal instabilitiesinannularcombustorshavenot beenabletosystem- aticallyexplore the full parameterspace provided by Eq.(5)due to constantly changing state space parameters [14] and statisti- cal preferenceforcertain parametercombinations[19].Therefore,

Fig. 4. All the 123 forced states on the Poincaré sphere, with 3D representation ( middle left ), projection on the plane χ= 0 ( middle right ) and projection on the θ= 0 plane ( right ). The relation between the parameters of the forced state and the coordinate system is illustrated on the left . The orientation angle θ0is restricted to the interval θ∈ [ 0 , π).

a forcing array is used to focus on the HRR for a fixed system state, expanding the limited combination of states that exist in self-excitedsystems.Forcing arrayshavebeenusedpreviously for limited combinations ofstate spacevariables in annularswirling combustors [19,47,64],andinthecurrentstudytheapplicationof the techniqueis improvedto obtainwell controlled forced states forthefullparameter spaceup tothephysicalamplitudelimit of theforcingarray.Similarlytothepreviousstudyin[47],twopairs ofhorndriversareusedtoimposetheforcedstate.Thetwohorn drivers ineachpairarelocated ondiametricallyopposite sidesof the combustion chamber, and are driven 180^◦ out of phase. The twopairsareseparatedbyanazimuthalangleof90^◦.

Thestrategyforimposingforcedstatesfollowsthesimpleidea ofmatchingtheamplitudeofbothspeakerpairs,eachsettingupa separate standingmode, andthenadjusting thephasing between the two pairs to obtain the desirednature angle

χ

^andorienta-

tionangle

θ

0.The forcingarray isapplied toathermoacoustically stableoperatingpoint toensuregoodcontrol ofthemode.While the mode is thermoacoustically stable, the preferencefor certain natureanglesobservedinself-excitedstudiesisstillprevalentdue to thereactingflow[19].Additionally,the combustorcanonlybe runforalimitedtimeduetothetemperaturelimitsoftheforcing array,meaningtotalthermalequilibriumcannotbeachieved.

Touseafixedfrequencyof f=1650Hzthecombustorisalways ignited atsimilar referenceouter wall temperature, andthedata acquisition is started approximately 20s later, ensuring the tem- peratures in the chamber are highly repeatable, and well-tuned totheforcing frequency.Thefrequencyisdeterminedbyiteration overdifferentfrequenciesatconstanthorndriverpower,choosing afrequencyclosetothemaximumresponseattheupperpressure transducerlocationwhileexhibitinggoodcontrolofthemode.The increasing temperatureof the combustionchamber afterignition introduces a slight drift in the mode from ignition to the time of data acquisition. However, the frequency is chosen such that thedriftisnegligibleduringthemeasurementperiod.Alivepres- suremodereconstructionisusedtoconstantlymonitorthemode, making itpossibletomakeadjustments totheforcing arrayuntil the measurements start. In general, standing forced states

χ

≈0 are veryeasy to obtainexperimentally,whilethespinning forced states

χ

≈ ±

π

/4arerelativelyhardtoobtainexperimentally.This difference ismostlikelycausedby combinationofthe preference forthestandingmodesduetonoise[42],andarequirementfora delicateamplitudebalancebetweenthespeakerpairsforthespin- ningmodes.

The final 123 prescribed forced states are indicated on the Poincaré sphere [18] in Fig. 4, and are summarised briefly in Table 1. The first azimuthal mode is the only one considered, and excited, in the current work. The maximum acoustic pres- surelevelisoftheorderof1%oftheoperatingatmosphericpres- sure, representativeofindustrialapplications[65].The numberof

Table 1

Overview of the range for the pressure mode parameters for the studied forced states. Close to the full parameter space is studied up to a finite amplitude limit imposed by the speaker array.

Parameter Range

Amplitude A in Pa 40 A 1100

Nature angle χ ^{−0 . 9 4} χ^/π ^{0 . 9}

Orientation angle θ0 0 . 0 θ0/π 1 . 0

forced states is more than 40 times higher than previous stud- ies insimilarconfigurations, andhighermagnitudenature angles areachieved[19,47]inadditiontoarangeofdifferentamplitudes.

Standing modes, as presentedin the lower part ofFig. 3, livein thehorizontalplanecuttingthecenterofthesphereinFig.4.The angle in the horizontal plane corresponds to the orientation an- gle

θ

0 ofthepressure anti-nodelocation. Converselythespinning modes,forexampleACWspinningmodeshownintheuppertplot of Fig. 3,live along the vertical axisin Fig. 4. Everything in be- tween are called mixedmodes, andcontain a mix ofa standing componentand a spinning component. Since the orientation an- gles

θ

0 and

θ

0+

π

^{of thestanding mode componentare equiva-}

lent, the range0−

π

^{is the main focus.The standingandmixed}

forcedstates are repeatedfor each

π

/4increment inorientation angleinthisinterval,ensuringeachinjector issubjectedtodiffer- entpointsinthestandingcomponentofEq.(5).Thefinalnumber of forcedstates comes froma desire to haveat least5 different amplitudesforthe4uniquenodallinepositionsforallthestand- ingandmixedmodenatureangles,aswellasdifferentamplitudes forthespinningmodes. Thisisthefirsttime thislevelofcontrol andsystematic exploration of the state spacehas been achieved forannularcombustorswithswirl.

2.5. Phaseaveraging

Turbulent fluctuations play a significant role in the instanta- neous heat release rate q, complicating the interpretation of the heatreleaseratefluctuationstructures.Inthefollowing,insteadof considering a real valued, time dependent function q(^r,

θ

,t)^{, the} respective analytic signal qa(^r,

θ

,t) ^{will be} considered. The ana- lyticsignalisdefinedasqa(^r,

θ

,t)=q(^r,

θ

,t)+H[q(^r,

θ

,t)^],where H istheHilbert transform.The analytic heat releaserateexpres- sion is then givenby the spatially dependent mean heat release rateq¯(^r,

θ

)^{,thephasedependentfluctuatingcomponentq}a(^r,

θ

,t) andthestochasticfluctuatingpartq_s,a(^r,

θ

,t)

qa

(

^r,

θ

^,t

)

^=q¯

(

^r,

θ )

^+qa

(

^r,

θ

^,t

)

^+qs,a

(

^r,

θ

^,t

)

^{. (8)}

Here the subscript “a” denotes it is the complex valuedanalytic expression ofthe fluctuations. The phase dependent fluctuations, whichareperiodicintimet,canbeobtainedfromphaseaveraging

theresponse q_a

(

^r,

θ

^,t0

)

⁼ _M¹

M−1

m=0

qa

(

^r,

θ

^,t0+mT

)

^−q¯

(

^r,

θ )

^{, (9)}

forasufficientlylargenumberofsamplesM,witht₀∈[0,T)^where T istheoscillationperiod.Thisisduetothestochasticfluctuations havingazeromeanforsufficientlymanysamples

M→∞lim 1 M

M−1

m=0

q_s,a

(

^r,

θ

^,t0+mT

)

^{=0. (10)}

The oscillation cycleis divided into 36 equally wide phase bins, resultinginaminimumnumberofsamplesofatleastM≈550in each bin,which is considered sufficient. Eachsample is then in- cluded inthephase averagebinclosest tothephase providedby theupperpressuretransducer(x=xupper)atthe

θ

=0location,as definedinFig.1.

2.6. RotationalaveragingandBlochtheory

One defining feature ofthisforced studyisthe useof N=12 injectorsarrangedequidistantlyaroundtheannulus.Thismakesit a primecandidatetousetheconceptofBloch theory[54],which wasrecentlyintroducedforsimulations ofannularcombustorsby Mensah and Moeck [8]. Mensah and Moeck used this to reduce thecomputationaldomainbyexploitingtheN foldsymmetryofa combustorwithN sectorstoonlycalculatetheresponseofasin- gle sector. However, in the currentstudy the response ofall the flames aremeasured,andBlochtheoryisinsteadusedtofindthe responseofanaverageflame,whichcorrespondstotheBlochwave part ofthe response.This willbe done in termsof theso called rotational averaging procedure first introduced in [47]. While the exactsamerotationalaveragingprocedureusedin[47]isfollowed here, this will nowbe interpreted in terms ofBloch theory. It is useful todescribe it in such terms,both because thenotation of the process can be defined more precisely, and also because of thecurrentpaper’sfocusontheflametoflamedifferences,which can be defined explicitly using this formulation. It also provides a useful experimental reference for a method which has so far only been applied numerically in the context of thermoacoustic instabilities.

2.6.1. Temporalmeanheatreleaserate

Thebaseideaoftherotationaveragingprocedureistoaverage all N=12 distinct flames together,to createan average flame. In the simplest casethis can be used to obtain the temporal mean HRR oftheaverage flamefromthetemporal meanflames q¯(^r,

θ

) through

^q¯

(

^r,

θ )

sectors= 1 N

N−1

l=0

¯

q

(

^r,

θ

+2

π

^l/N

)

, (11)

where theN fold rotational symmetryis exploitedto add allthe distinct flames together to getan average responseandno Bloch theoryisrequired.Thisreducesthespatialdependencetoasingle flamesector.ThetemporalmeanHRRofthedistinctflamesq¯(^r,

θ

) and the corresponding temporal meanHRR of the averageflame

^q¯(^r,

θ

)

sectorsareshown for thechosen operating condition,which is thermoacoustically stable, foran unforced caseinFig. 5.Over- all, the main flame structures are observed to be similar forthe two quantities, butthereare also some differencesinthe spatial distribution of theheat release ratefrom flameto flame. Allthe flames exhibit both positive and negative differences simultane- ously, meaningthenetresult,shownonthebottomright,arerel- ativelysmallbutshouldstill beconsidered.Therefore,inthecase ofexaminingthefirstazimuthalwaveresponseoftheheatrelease

Fig. 5. Temporal mean of the distinct flames ( upper left ) for a stable, unforced case and the corresponding temporal mean of the average flame ( upper right ) without flame to flame differences. The flame to flame differences are quantified ( lower left ) by subtracting the average flame ( upper right ) from the distinct flames ( upper left ).

The net differences ( lower right ), obtained by sector averaging, are presented as col- ored regions. The structure is observed to be similar for the distinct and average flames, but there are some significant differences in the spatial distribution of the HRR within the individual flames. However, the differences in all the flames are both positive and negative, resulting in a much smaller net difference.

ratefluctuations,thefluctuationsshouldbenormalisedagainstthe corresponding average flame temporal mean HRR

^q¯(^r,

θ

)

sectors, andnotthetemporalmeanofthedistinctflamesq¯(^r,

θ

)^.

2.6.2. Heatreleaseratefluctuations

Thesamefundamentalideaofhowtoobtainanaverageflame responseisalsousedtointroducetherotationaveragingprocedure forthefluctuatingpartoftheheatreleaserate.Thebaseassump- tionwillbethat theflamesarethesameandtheonlysignificant azimuthalcomponentinthemeasuredresponseisthefirst,degen- erate,azimuthal mode. This isreasonablebecause the prescribed forcedstate is restrictedto the first azimuthal mode. The flames willalsobe modeled asrespondingonlyattheforcing frequency f=

ω

/2

π

^{,becauseinFig.2theamplitudeoftheharmonicsareat}

leasttwo orders ofmagnitudelower forall cases.UtilizingBloch theory,theanalyticalphaseaveragedheatreleaserateinEq.(9)is modeledas

q_a

(

^r,

θ

^,t

)

⁼

ψ

−1

(

^r,

θ )

^e−iθ+

ψ

+1

(

^r,

θ )

^{eiθ +}

^qˆa

(

^r,

θ )

e^iωt. (12) The functions

ψ

±1(^r,

θ

) ^{are the spatially dependent heat release} ratemodeshapes,whicharecalledBlochkernels.Thecorrespond- ingcomponentsinEq.(12)accountforthedegeneracyofthefirst azimuthalmode.

ψ

− istheamplitudeoftheACWrotatingcompo- nent,and

ψ

+istheamplitudeoftheCWrotatingcomponent.The flametoflamedifferences,thatarepresentinallphysicalsystems, areaccountedforinthe^qˆaterm.Thesedifferencesareassumed tobe small,madeformally explicitby the inclusionof thefactor 0<

1intheexpression.The^qˆatermingeneralaccountsfor theviolationofthemainassumptionsmadebeforeEq.(12),mak- ing it possible to describe an arbitrary response.Eq. (12) is also abletoaccountforflame-flameinteractions,assumingthatthein- teractionisthesamebetweenthedifferentflames.

TheBloch kernels

ψ

±1 aredefinedonasingleflamesector,as definedinFig.1,andareperiodic[8]:

ψ

±1

(

^r,

θ

⁺²

π

^/N

)

⁼

ψ

±1

(

^r,

θ )

^{, (13a)}

^qˆa

(

^r,

θ

⁺²

π )

⁼

^qˆa

(

^r,

θ )

^{. (13b)}

Inspired by Eq.(11) therotational averageintheACW (-)and CW(+)directionsareintroducedasthesum

q_a

(

^r,

θ

^,t0

)

±

= 1 N

N−1

l=0

q_a

(

^r,

θ

^∓2

π

^l/N,t0+2

π

^l/

( ω

^N

) )

⁽¹⁴⁾

corresponding to rotatingthe coordinatesysteman angle∓2

π

/N foreachstepintime2

π

/(

ω

^N)^{.Thisdefinitionisexactlythesame} asoriginally proposed in[47].Theleft handside willbe referred to asthe ACW and CW rotation average componentsfor negative and positive sign respectively. It can be shown that the rotation averagecomponentsinEq.(14)areequivalenttothecorresponding azimuthalwavecomponentsdefinedbytheBlochkernels

ψ

±1

q_a

(

^r,

θ

^,t0

)

±

=

ψ

±1

(

^r,

θ )

^{e±iθeiωt0. (15)}

TheproofofthisispresentedinAppendixAforcompleteness.The phase ofthe componentinEq.(15)is determinedfromthe tem- poralphase

ω

^t0selectedforthefirststepintherotationaveraging process.Eq.(14)canalsobeusedtoobtainhigherspatialharmon- icsoftheHRRbychangingtherelationbetweentherotationterm andthe timestep.In thecurrentstudythehigherharmonicsare ofnegligibleorder,andwillnotbeconsidered[66].Thetechnique canalsobeslightlymodifiedtoworkwithdifferentinjector types inthe sameannularcombustor, aslongasthedistribution ofthe different injectorsiscyclicwithmorethan asingleperiodinthe azimuthaldirection.

Anotherwaytoconsider thisprocessisasfollows.Ifthetotal phaseaverageresponsecorrespondstooscillationswhichtravelin both CWandACWdirections aroundthe annulusonceper cycle, each different flameresponds withadelayin phase totheseos- cillations.Therotationalaverageessentiallytravelswiththeoscilla- tions;averagingtogether theresponseofeachindividual flameat thesamephaseintheoscillationcycle.AstherearebothACWand CW oscillations, two rotationalaverages canbe calculated,yield- ingtheaverageresponseofallflamestoeachphaseintheoscilla- tioncycle.Inpracticethephaseaveragewasdividedinto3N=36 equallyspacedphasebins,whichcanbethoughtofas3individual time series wherethe bins within each individual seriesis sepa- ratedby^t=2

π

/(

ω

^N)^{.Thestart timet}0 ofthe3seriesaresep- arated bya timedeltaof2

π

/(³

ω

^N)^{,providing3timesbetter in-} terpolatedtemporalresolutioncomparedtodividingitintoN=12 bins.Foreachofthe3timeseriestherotationaverageinEq.(14)is calculatedfortherealvaluedphaseaverageimageswherethero- tationsareperformedbyphysicallyrotatingthecoordinatesystem oftheimagesforeachtimestep.

2.6.3. Reconstructedheatreleaseratefluctuations

TherotationaveragedcomponentsinEq.(15)aretheBlochaz- imuthal wavecomponentsineach directionofthephase average, and are by definitiondescribing the average flameresponse to a first azimuthal mode. This can be utilized to recreate the phase average inEq.(12)withouttheflame toflamedifferences(

^qa) byperformingtheoppositerotation¹

q_a^,rec

(

^r,

θ

,tl

)

⁼

q_a

(

^r,

θ

+2

π

l/N,t0

)

+

+

q_a

(

^r,

θ

−2

π

l/N,t0

)

−

, (16)

wheret_l=t0+2

π

^l/(

ω

^N)^{.Theresponseofthe}reconstructedphase average effectivelyeliminatesflametoflamedifferencesfromthe phase averageresponse foreachforcedstate, enablingclearerin- terpretationofstructuresanddynamics.Inpracticaltermsthereal partof thereconstruction isobtainedasthesuperpositionofthe

1Equation (12) without flame to flame differences q ais retrieved by inserting Eq. (15) into Eq. (16) .

two real valued rotation averages rotated a fixed angle in oppo- sitedirections,wherethe angle±2

π

^l/Ncorresponds totime step t_l in the phase average. The same method can be used to re- construct the time series of the individual Bloch wave compo- nents

ψ

±1e^i(±θ+ωt)

by only including one of the right hand sidetermsinEq.(16)withoutconvertingtofrequencyspace.

2.6.4. VelocityfluctuationsandtheAzimuthalFDF

Thesamerotationalaveragingprocedureandreconstructioncan alsobeappliedtotheaxialvelocityperturbationsevaluatedatthe dumpplane.Thisassumesthattheaxialvelocityperturbationscan beexpressed asBloch wavescorresponding tothefirstazimuthal mode. Since the induced acoustic mode in the injector tube is predominantlydeterminedthroughpressurecouplingwiththeaz- imuthal pressure mode[59],the induced axial velocity perturba- tions,consideredattheinjectorcentrelocation,aredeterminedby thelocalazimuthalpressuremodeamplitude.Forbrevitytheiden- ticalderivationwill not beshown, butthe resultsare equivalent.

Since theaxial velocity perturbationsare calculated forthecom- plex valued Fourier amplitudes by default, the rotation averaged componentoftheaxialvelocityamplitudeatthedumpplane can beobtainedfrom

uˆ_axial

θ

j

±

= 1 N^∗

N^∗−1

l=0

ˆ u_axial

θ

j∓2

π

^l/N∗

e^i2πl/N∗. (17)

Here the N^∗ instrumented injectors are assumed to be equidis- tantlyspacedintheazimuthaldirection,meaningN^∗=N/2=6in thecurrentconfiguration.Eq.(17)describestheaxialvelocityper- turbations astwo counter propagating azimuthal waves,and the lefthandsideisthereforedenotedanazimuthalaxial velocityper- turbation component. In a similar manner as Eq.(16) the recon- structed axial velocity Fourieramplitude can simply be obtained from

uˆ_axial^,rec

θ

j

=

uˆ_axial

θ

j

+

+

uˆ_axial

θ

j

−

. (18)

As a shorthand, andto be morein linewith thenotation ofthe sectoraverages

(·)

j,thefollowingnotationalconventionisintro- duced

uˆ_axial

± j =

uˆ_axial

θ

j

±

, (19a)

ˆ

u_axial^,rec_,j=uˆ_axial^,rec

θ

j

. (19b)

Oneadditionalmajoradvantageofthevelocitysignalsasshown inEq.(18)isthattheaxialvelocityperturbationsareobtainedfor allthe injectors.Thisdoubles theamount ofusabledata,asonly every other injector is instrumenteddue tothe cost ofacquiring pressuresignalsandtherequiredstorage.Itisthennaturaltode- finetheAzimuthalFlameDescribingFunctionas

FDF^±

ˆ u_axial

±

= 1 N

N−1

j=0

qˆ

±

j/

^q¯

sectors

j

uˆ_axial

± j/Ubulk

, (20)

wherethespatialaveragingisagainperformedoveroneflamesec- tor, as defined in Fig. 1. The domain of the functions

ˆ q

±

and

^q¯

sectors isa singleflamesector, howeverduetoimperfectionsin centeringofthe flamesector masksandsmalldifferencescaused bydefiningthemasksonasquaregrid,theAzimuthalFDFiscal- culatedasthemeanofallNflamesectorstoreduceeffectsofthis.

2.7. Azimuthalheatreleaseratemode

Inthe previous section the Azimuthal Flame DescribingFunc- tionwasintroduced.Thiscanbeusedtodirectlyexpressthenor- malised,spatiallysectorintegratedheatreleaseratefluctuationsof