Combining analytical models and LES data to determine the transfer function from swirled premixed flames

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Dupuy, Fabien and Gatti, Marco and Mirat, Clément and

Gicquel, Laurent and Nicoud, Franck and Schuller, Thierry

Combining analytical models and LES data to determine the

transfer function from swirled premixed flames. (2020)

Combustion and Flame, 217. 222-236. ISSN 0010-2180

Official URL:

https://doi.org/10.1016/j.combustflame.2020.03.026

Combining analytical models and LES data to determine the transfer

function from swirled premixed flames

Fabien Dupuy

a,b,_.,

Marco Gatti

_C,

Clément Mirac

c

, Laurent Gicquel

a_,

Franck Ni coud

e

,

Thierry Schuller

d

• CERFACS, 42 Avenue Gaspard Coriolis 31057 Toulouse Cedex 01, Fronce b Safran Aircrafr Engines, Rond-poinr Rene Ravaud, Moissy-Cramayel 77550, France

<tabomroire EM2C. CNRS, CenrmleSupélec, Universiré Paris-Saday, 8-10 rue Jolior Curie 91192 Gif Sur Yverre cedex, France d/nsrirur de Mécanique des Fluides de Toulouse, IMFT. Universiré de Toulouse, CNRS, Toulouse, France

• fMAG, Univ. Monrpellier, CNRS, Monrpellier, Fronce

Keywords:

Flame Transfer Function Swirling flame

large Eddy simulation Analyrical model

1. Introduction

ABSTRACT

A methodology is developed where the acoustic response of a swirl stabilized flame is obtained from a reduced set of simulations. Building upon previous analytical flame transfer functions, a parametriza­ tion of the flame response is first proposed, based on six independent physical parameters: a Strouhal number, the mean flame angle with respect to the main flow direction, the vortical structures convection spe.ed, a swirl intensity parameter, a time delay between acoustic and vortical perturbations, as well as a phase shift between bulk and local velocity signais. lt is then shown how these parameters can be de­ duced from steady and unsteady simulations. The methodology is applied to a laboratory scale premixed swirl stabilized flame exhibiting features representative of real aero-engines. ln this matter, cold and re­ active flow Large Eddy Simulations are first validated by comparing results with reference data from experiments. The high fidelity simulations are seen to be able to capture the flame structure and velocity profiles at different locations while forced flame dynamics for the frequency range of interest also match the experimental data. From the same analytical transfer function model, three methodologies of increas­ ing complexity are presented for the determination of the model parameters, depending on the available data or computational resources. A first estimation of the flame acoustic response is obtained by evalu­ ating parameters from a single stationary flame simulation in conjunction with analytical estimations for the acoustic-convective time delay. Flame dynamics and swirl related parameters can then be determined from a series of robust treatments on pulsed simulations data to improve the mode! accuracy. It is shown that good qualitative agreement for the flame transfer function can be obtained from a single non-forced simulation while quantitative agreement over the frequency range of interest can be obtained using ad­ ditional reactive or non-reactive pulsed simulations at one single forcing frequency corresponding to a local gain minimum. The method also naturally handles different perturbation levels.

Increasingly stringent regulations for pollutant emissions have

lead most aeronautical engine manufacturers to aim for systems

using lean premixed combustion as an efficient way to reduce NOx

while maintaining a good level of performance. However, lean pre­

mixed combustors have been shown to be prone to combustion in­

stabilities where the coupling between acoustics of the combustor

and the flame can cause structural damages to the engine

[1 ). In

parallel to these engine developments, considerable numerical ef­

forts have been deployed and Large Eddy Simulations (LES) for in­

stance has proven capable of capturing combustion instabilities in

complex geometries [2,3). Given the broad range of operating con­

ditions of an aircraft engine, and its intrinsic complexity, a system­

atic use of high fidelity simulations is however still out of reach at

the design stage. For this reason, less numerically intensive meth­

ods have been developed in the form of low order codes for ther­

moacoustic instabilities where acoustics and the complex flame

dynamics are decoupled.

• Corresponding author ar: CERFACS, 42 Avenue Gaspard Coriolis 31057 Toulouse Cedex 01, France.

E-mail addresses: fabien.dupuy@cerfacs.fr. fabien93.dupuy@orange.fr (F. Dupuy).

A large majority of such codes solve the Helmholtz equation

in the frequency domain and take into account the flame or the

associated unsteady heat release response to acoustic perturba-tions using a Flame Transfer Function (FTF) [4] or more recently a Flame DescribingFunction (FDF)to also account fornon-linear effects [5,6]. This approach allows to perform parametric stud-ies, hence enabling to tackle combustion instabilities issues dur-ingthedesignphase[7,8].Examplesofresultsobtainedwithflow solvers using the Finite ElementMethod (FEM)to determine the solutionofthethree-dimensionalHelmholtzequationwithin com-plex geometries withcomplex boundaryconditions,including ac-tive flames and/or eventually also damping or nonlinear effects become numerous [9–14].These methods rely on the use ofthe FTF/FDF and its determination is therefore of crucial importance if one wants to control or even predict thermoacoustic instabili-ties. Restricting the analysisto swirled flames,the determination of FTF/FDF can be made experimentally [15–17] as well as with numericalflowsimulations[18–22].Severalanalyticalformulations basedonalevelsetapproacharealsoavailable,butonlyafraction deals with swirling flames. You et al. [23] derived a FTF model based on a triple decompositiontechniqueof the G-equation in-cludingeffectsofthemean,synchronizedandstochasticflow vari-ables to deal withflow rate disturbances andalso changes asso-ciatedtoequivalenceratioperturbations.Similarapproacheswere usedin[24,25] toexaminetheFTFofturbulentpartiallypremixed flamesofarbitraryshapeduetoequivalenceratioperturbations.

Ingasturbines,flamesarehighlyturbulentandstabilizedbya swirlingflow. The injector impartsarotating motiontothe fresh gases,sothat combustionoccursaroundaninnerhot gas recircu-lation zone that helps anchoring theflame in the vicinity ofthe injection system. The Swirl number, S, is generallyused to char-acterize therateofrotation oftheflow andisdefinedasthe ra-tiobetweenthetangentialandaxialmomentumfluxes.Neglecting pressuretermsasoftendoneintheliterature,itreads:

S=1_R R 0

ρ

uzu_θr2dr R 0

ρ

u2zrdr (1) where

ρ

isthefluid density,uz andu_θ arerespectively theaxial and tangentialvelocity components using cylindricalcoordinates, and R is a characteristic dimension, usually the outer radius of the injection device. Recent studies [15,26,27] on swirling flame dynamics indicatethat their responseis not onlydictatedby the flame itself, butalsoby the swirlerresponse as evidently shown byvarying theswirlertocombustionchamber backplanedistance or the bulk flow velocity in the injector.Palies etal. [28] there-foreproposed ananalytical FTFmodelforinvertedconicalflames (or V-shaped)swirledflames basedonthe perturbed G-equation. Inthiscase, theswirlactionischaracterized byalinear relation-ship between the normalizedaxial acoustic velocity perturbation and the tangential convective velocity perturbation generated by acoustic-vorticity conversion in theswirler [15,29]. This formula-tion offers an interesting insight on the mechanisms driving the response of flames subject to swirl perturbations. Note that the model in [28] uses the convectively perturbed V-flame FTF from

[30] as a basis and was shownto be in good agreement with ex-perimentsforthefewconsideredcases.

The presentstudy extendsthe work fromPalieset al.[28] to properlyrepresentV-shapedswirlstabilizedflames.Moreover, nu-merical simulations areused todeterminethe FTFmodel param-eters instead ofrelying onexperiments. Following thisapproach, it is shownhow toobtain the flame responsefor a whole range offrequencies witha minimumsetofsimulations ofan unforced flame ina thermo-acoustically stable state. As a complement, an increaseinthemodelfidelityisdemonstratedbyaddressinga re-ducedsetofacousticallyforcedsimulationsinadditiontothe sta-blesteadycase.Thiswouldconstituteaconsiderableimprovement fortheFTFdeterminationoverclassicalmethodssuchasusing

nu-merousforcedLES, orevenmoreadvanced techniquesrelying on systemidentification [31] which are limited to vanishingly small perturbations.

In the following, a description of the model forthe FTF of a swirledV-shapedflameisfirstpresentedinSection 2,theroleof eachparameterbeingdiscussed.Theleanpremixedswirledflame configuration is then presented in Section 3. LES predictions are validated against experimental data for cold and reactive condi-tions inSection 4, followedby a first estimationof the FTF.It is shown that qualitative agreement for the FTF gain and phase is obtainedwhenextractingparametersfromanunforcedflame sim-ulation.Finally,inSection5,flamedynamicsasobtainedin acous-tically forced LES are validated against experimental results and moreaccurate evaluationmethods for model parameters estima-tions are proposed. Model andexperimental flame responses are finallycompared,demonstratingtheabilityofthemethodologyto predicttheFTFgainandphasewithagoodaccuracy.

2. Flametransferfunctionmodelingandtheory

Manyanalytical expressions forthe FTF based on a lineariza-tion of the G-equation have been derived since the pioneering work fromFleifiletal.[32] forthe frequencyresponse ofa con-icallaminarpremixedaxisymmetricflamestabilizedonatuberim whichissubmitted toharmonicflow-rate modulations.Assuming thattheflameisa thininterfaceseparatingfreshandburntgases andmoving atthelaminarflamespeeds_l inthedirectionnormal toits surface, whatis generallydescribedbya G-equation,Fleifil etal.[32] wereabletoderiveananalyticalexpressionforan elon-gatedconicalflameprimarilydependingonaflameStrouhal num-ber. The vast majority of analytical FTF models in the literature wasthen also derived froman analysis involving a perturbed G equationwhetheritbeinagenerallaboratoryframeorinaframe linked to the mean flame front position for conical orV-shaped flames.Inthefollowing,focusismadeonV-shapedflameslikethe onedescribedinFig.1.Localflamewrinklingisassumedtooccur aroundthismean positionwhichdoesnot changewhen acoustic forcing is applied,which translatesfor a premixedflame to only consideringlinearacoustics[33].

Forthe model derivation, although alternatives exist [34], we make use here of the convective model for laminar V-shaped flames introduced bySchuller etal.in [30] andwherea convec-tivevelocityperturbationpropagatesintheverticaldirectioninthe laboratoryreferenceframe.Thatis:

v

=

v

1exp



i

ω

v

y− i

ω

t



=

v

1exp

(

iky− i

ω

t

)

(2)

where k=

ω

/

v

is a convective wavenumber builtfrom the aver-ageaxial flow speed

v

and the angular frequency

ω

. Parameters

a and b in Fig. 1 correspond to the flame anchoring point and theradialflameextensionrespectively.Theanalysisisrestrictedto swirledflamesanchoredonanarrowstabilizingrod,sothata b. Integratingtheunsteady flamedisplacementover theflamefront andconsideringthattheunsteadyheatreleaserateisdirectly pro-portional to the unsteady flame surface variation, Schuller et al.

[30]obtaintheanalyticalFTFreproducedbelowforconvenience: Fv=

_ω

22 ∗ 1 1− cos2

α



eiω∗_{− 1−}e iω∗cos2α− 1 cos2

α



+

_ω

2i ∗ 1 1− cos2

α



eiω∗cos2α_{− e}iω∗ ₍₃₎

where

ω

_∗=

(

ω

Rf

)

/

(

slcos

α

)

=

(

ω

L2f

)

/

(

v

Hf

)

=

(

ω

Lf

)

/

(

v

cos

α

)

is a non-dimensionalreduced frequencyrelatedtothe flameStrouhal number of Fleifil et al. [32] and

α

=sin−1

(

sl/

v

)

=cos−1 Hf/Lf isa flameaspect ratioparameteror simplytheflame anglewith respect to the main flow direction, relatedto the laminar flame

Fig. 1. Schematic of the studied V-shaped flame configuration. Reference frame axes are noted x and y while a second frame directly linked to the steady flame has axes X and Y . The steady flame is aligned on the X axis. It is anchored on a rod at position a and extends to a radial abscissa x = b which may not be a wall. Definitions of the flame length L f , flame height H f and flame radius R f are provided on the right (adapted from [30] ).

speedsl.The ratio

ω

∗/

ω

representsthe convectiontime fromthe anchoringpoint to theextremity oftheflame sheetatthe mean flow velocity. This first expression solely depends on mean flow and geometrical quantities, which is not surprising in the con-text of laminarflames it was originally developed for, butraises questionsforturbulentflameswheredynamicsarelikelytobe of crucial importance. As pointed out by Preetham et al. in [35], a limitationof this modelstems fromthe fact that the convective velocityin Eq.(2) assumesthat perturbations travel atthe mean flowvelocity

v

.Inrealitytheseperturbations(thatwewill assimi-latetovorticalperturbations tosimplifytheanalysis)travelinthe outer shear layer in the case of a V-shaped flame at a velocity

Uc−v that maybe lower than

v

[36] but cannot exceed it.

Intro-ducingthetruetomeanvelocityratioK,onethereforenecessarily hasK= v

Uc−v ≥ 1.Eq.(2) canaccordinglybemodifiedbyaddingthe correctionfactorKtothewavenumber,whichyieldsthefollowing expressionfortheFTF:

Fv=

_ω

22 ∗ 1 1− Kcos2

α



eiω∗_{− 1−}e iω∗Kcos2_α − 1 Kcos2

α



+ 2i

ω

∗ 1 1− Kcos2

α



eiω∗Kcos2_α − eiω∗ ₍₄₎

Apotentialdrawbackforthemodelasis,especiallyforwideflame angles,isthatitmayresultinlarge gainvaluesforhigh frequen-cieswhicharenotobservedinexperimentsforthistypeofflames

[37].IndeedSchulleretal.[38] show byuseofPIVmeasurements in the fresh gases that the velocity perturbation amplitude de-creaseswiththeaxialdistance,withafrequencydependentdecay rate. The authors also pointedout that thisfeature is neededto retrievetheFTFobtainedexperimentallywhenusingamodel de-rivedfromaG-equation.Furtherstudiesfocusingonthefresh re-actantsside[39] foundvelocityperturbationstohavean exponen-tialdecayratewhichincreaseswithfrequency.Byconstruction,the FTFmodel _Fv only considers one-dimensional propagation with-outanydecay,whichmaybetrueintheinjectionsystembutmay nothold astheperturbation goesthrough the largercombustion chamber enclosure. To take this feature into account, the spatial component

v

ˆ ofthe convective velocity

v

=

v

ˆe−iωt _{in Eq.(2) can} againbemodifiedfollowingtheformulationproposedin[40]: ˆ

v

=

v

1exp



iK

ω

v

y



exp



−

β

K

_v

ω

y



=

v

1exp



iK

(

1+i

β

)

ω

v

y (5)

whicheffectivelycomesdowntointroducinganewcomplex veloc-itycorrectionfactorK=K

(

1+i

β

)

insteadofareal-valued quan-tity.Inthiscase,thecorrespondingdecayrate,−

β

K

ω

/

v

,increases with frequency which complies with experimental findings. This means that Eq.(4) retains the same form, andin the latter, the parameterKisalwaysassumedtobecomplexforconciseness pur-poses.

Tothis point, _Fv holds for a laminarV-shaped flame without anyconsiderationregarding swirloranyazimuthal velocity com-ponent.Analyticalexpressions forswirlingflamesare muchmore scarcethanforstandardlaminarflames.Themodelpresentedhere isaslightlymodifiedversionoftheexpressionintroducedbyPalies etal.[28] forswirlingV-shapedflames.Ithasalreadybeenapplied withsomesuccessin[41] togaininsightonthedynamicsof strat-ifiedswirlingflames.Paliesetal.[28] showthattheswirlingflame FTFcanbewrittenas:

Fs=Fv



1−sˆt/st ˆ

v

/

v



(6) whereFv isthe laminarV-shaped flameFTF that waspreviously

detailed andst is the turbulent flame speed. According to Palies etal.[28],turbulentvelocityfluctuationsareassumedtobelinked to the normalized velocity fluctuations in a linear fashion using tworealvaluedmodelparameters

χ

and

ζ

sothat:

ˆ st st =

χ

ˆ u_θ u_θ +

ζ

ˆ

v

v

, (7)

The axial

v

ˆ and azimuthal uˆ_θ velocity disturbances are then as-sumedtoberelatedby:

ˆ u_θ u_θ = ˆ

v

v

e iφuˆ_θ−ˆv ₍₈₎

This means that the normalized velocity fluctuations are essen-tiallythesamebutwithaphaseshift

φ

uˆ_{θ −ˆ}vbetweentheaxialand

theazimuthal components.ReintroducingEqs. (7) and(8) inEq. (6)thenyieldstheFTFforaswirlingV-shapedflame:

Fs=Fv 1−



ζ

+

χ

eiφuˆ_θ−ˆv



(9) which dependson sixparameters :

ω

∗,

α

, K,

χ

,

ζ

and

φ

_uˆ

θ −ˆv.In the original work ofPalies etal.[28],no additional constraintis present. Onecanhoweverpoint outthat accordingtotheory,the FTF gain at zero frequency should always be unity [42]. Setting

φ

uˆ_{θ −ˆ}v=

ωτ

where

τ

is a characteristictime delaybetween axial

χ

=−

ζ

.Theothersolution

χ

=2−

ζ

isdiscardedasityieldsnon physicalresults.Thissimplisticassumptiondoeshowevernothold whenconfrontedtotheexperimentalfindingsfrom[28]. Introduc-ingamoregeneralframeworkbysetting

φ

_uˆ

θ −ˆv=

ωτ

+

φ

0and

en-forcingthelowfrequencylimitgainyields:

|

1−

ζ

+

χ

eiφ0

|

=_{1 (10)}

where|· |standsforthemodulusofacomplexnumber.Itresults asecondorderequationrelating

χ

and

ζ

:

ζ

2₊₂

ζ

₍

χ

_cos

φ0

_{− 1}

₎

₊

χ

2_{− 2}

χ

_cos

φ0

_{=0 (11)}

Assuming

χ

and

φ

0tobeknown,solutionsofthisequationare:

ζ

1=1−

χ

cos

φ

0+ 1−

χ

2sin2

φ

0 1 2 (12)

ζ

2=1−

χ

cos

φ

0− 1−

χ

2sin2

φ

0 1 2 (13) Assuming |

χ

| ≤ 1,

ζ

1 and

ζ

2 take real values.Without anydata

tocomparethemodelresults,anddependingonparametervalues, itisdifficultto choosebetweenone rootoranother. Experiments forV-shaped flames FTF haveshownthat such flames exhibit an increase ingaininthelow frequencylimit [37].StartingfromEq. (9)andinsertingEq.(12) or(13),onecanshowaftersomecalculus that to first order, the low frequency derivative for the FTF gain reads:

lim

ω→0

∂|

Fs

|

∂ω

=

τ

sin

φ

0

χ

2cos

φ

0±

χ

1−

χ

2sin2

φ

0

1 2



(14) Therefore,thevalue ensuringapositive derivativeoftheFTFgain in the low frequency limit is chosen depending on values for

χ

,

τ

and

φ

0. Eq. (9) along with Eq. (12) or (13) constitute the

parametrization of a V-shaped premixed swirling flame transfer function that will be referred to as the SFTF model throughout the restofthe article.Thismodel relieson a setofsix indepen-dent parameters,three ofwhichdescribe the premixedflame re-sponse(

ω

∗,

α

,andK)whilethethreeremainingonesaccountfor the effect of the swirling motion (

χ

,

τ

and

φ

0). While the

cur-rentmodelislimitedtopremixedV-shaped flames,itcould han-dlevariousflowinjectionconditionsandfuelsthroughparameters

ω

∗ and

α

whilevariousswirlerdesignswouldaffectparameters

τ

,

φ

0 and

χ

.Notethatan overviewoftheroleofeachparameteris

available insupplemental material A inthe form ofa sensitivity analysis. From this study, it is concluded that every investigated parametershouldbeevaluatedwithcaution,andemphasizesboth theneedforan accuratedescriptionofflamedynamicsaswell as forapreciseevaluationofintermediatequantitiesneededto eval-uate modelparameters. Thisevaluationprocess should ideallybe robustenoughtocoveralargespectrumofconfigurations.

3. Testconfigurationandmodeling

To proceed with the SFTF model validation, LES of a generic configurationareperformed.Inthismatter,alaboratoryscale pre-mixedswirlstabilized burner ischosen asitfeatures a turbulent swirlingflowandaneasytomodelgeometry.Thenumericalsetup isconceivedtobe robustenoughtoallowfocusontheevaluation ofquantitiesofinterestwithreducedrelativeuncertainties.Details regardingtheexperimental rigandthenumericalmodeling strat-egyarepresentedinthefollowingsection.

3.1. Experimentalsetup

Theexperimentalconfigurationusedforanalysisandmodel de-velopmentisavariationoftheNoiseDynburneralreadystudiedin

[43–46].Therigiscomposedofaninjectionsystem,aswirlerunit

Fig. 2. Sketch of the NoiseDyn swirl combustor, with dimensions in mm. Only the shaded domain is resolved in the LES, starting 6 mm under the hot wire position (HW). δis the distance between the combustion chamber backplane and the top of the swirler channels exit.

anda combustion chamber ending with a short exhausttube as presentedinFig.2.Apremixedmethane/airmixtureofequivalence ratio

φ

₌0_.82 is injected through the bottom of the device and passesthroughahoneycomblayertobreaklargeturbulenteddies, the systembeing operated atP0=1 atm andwith a freshgases

temperature T0=293 K. The resulting laminar flow enters a 65

mmwideplenumandaconvergentofdiameterDin=22mm, pro-ducinga top-hatvelocityprofilewithbulkvelocityUb=5.44m/s atthehot wirepositionidentified asHW inFig.2.Theflow then goesintoaradialswirlerusingsixcylindricalchannelsofdiameter

dSw=6mmforminga33◦ anglewithrespecttotheradial direc-tion,beforeleavingthroughacentral20mmwideinjectiontube. Theinjectionalsoincludesacentralmetallicrodofdiameterd₌6 mmtoppedbya10mmhightruncatedconeendingwithacircular sectionofdiameterDc=10mmtohelpstabilizingtheflame.Note thatthe coneisprotruding 1.5mm insidethecombustion cham-berandthechamber back-planetoswirlerback-plane distanceis

Lbc=56 mm. The combustion chamber itself is 150 mm long, is equippedwithfourquartzwindowsandhasasquarecross-section ofL_ch=82mmwidth.Intheabsenceofacousticforcing,theflame isanchoredfewmillimetersabovetheconeandhasaVshape.

This academic configuration is not as complex as injectors used in real engines, yet it includes their most prominent fea-tures with a turbulent swirling flow and confined flame oper-ated at lean premixed conditions. It allows for an easier inter-pretation of LES results than in real scale burners. A wide se-lection of diagnostics has been employed by Gatti et al. [47,48], includingvelocity measurements witha hot wire,pressure mea-surementsusingmicrophones,LDV andPIV oftheflowatthe in-jection exit and in the flame residence area. FDF data is avail-able for forcing amplitudes ranging from 10 to 70% RMS of the bulk velocity at thehot wirelocation. In the following, only the unforced data and the one at a forcing amplitude

v

_RMS=30% is considered.

Table 1

List of meshes created during the adaptation process and their characteristic dimen- sion h in the swirler, injector and flame regions.

Mesh id. hswirler( mm ) hin jector( mm ) hf lame( mm ) number of tetrahedra

M1 0.306 0.379 0.515 15 526 871 M2 0.253 0.327 0.515 16 234 106 M3 0.251 0.290 0.476 19 110 948 M4 0.158 0.197 0.260 55 766 260

3.2.Numericalmodeling

LES ofthe configuration described in the previous subsection areperformedusingtheAVBPsolverdevelopedbyCERFACS,which solves the three-dimensional filtered compressible multi-species Navier–Stokesequationsonunstructuredgrids.TheTTGCcentered spatial scheme [49] is used, featuring a third order accuracy in both spaceand time. The computational domain covers a region startingafter the plenum contraction all the way to the exit of theexhausttubetoreduce thetotal costofthesimulation.Since the flame response to acoustics is to be studied, characteristic boundary conditions[50] are used for both inlet and outlet, en-suring a proper treatment of waves. All other boundary condi-tions are set to adiabaticno slip wallsfor the cold flow simula-tions,whileforhot conditions,chamber wallsaswellasthe con-icalpart of the stabilizingrod are specified asheat losing walls. Todoso,experimental temperaturemeasurementsoutsideofthe quartz panels are used as a referencealong withadequate ther-malresistances,allowingwalltemperaturesintheLEStoadaptto thechamber temperaturefield. Theconicalbluff-bodyisassumed tohaveaconstanttemperatureof293Katitsbaseinsidethe in-jectionunit andthe fin theory is used to derive a space depen-dent analytical expression for the thermal resistance. Since few experimental data is available for the combustion chamber back planetemperatures, ahyperbolic tangent profilevarying between 300Kinthecentral watercooledregionto700Knearthe cham-berwallsisprescribed. Finally,theSIGMA subgridscalemodel is used[51] whileflame/turbulenceinteractionishandledusingthe DynamicallyThickenedFlamemodel(DTFLES[52]) inconjunction with a two-step BFER chemistry [53] validated for atmospheric conditions.

Formeshing,themeshadaptationstrategyproposedbyDaviller etal.[54] isemployed,resultinginthreemeshesM1,M2,M3 pre-sentedinTable1,wherehisthe characteristiccellsize ina par-ticularmesharea.

To proceed, a baseline full tetrahedraunstructured mesh, M1, wasfirst created,withrefined regions aroundtheswirler,the in-jectionsystemandthe area wherethe flame issupposed to sta-bilize. The normalized time averaged viscous dissipation defined by:

=

(

μ

+

μ

t

)



∂

ui

∂

xj +

∂

uj

∂

xi 2 (15) where

μ

and

μ

t are thelaminar andturbulent dynamic viscosi-tiesrespectivelywasthen extractedfromtheassociated LES pre-dictionsandused asametricfortheautomatic meshrefinement process.Itresultsanewmesh,M2,fromwhichtheprocessis iter-atedonceagaintoyieldmeshM3.Thismethodallowsnotonlyto adaptthemesh nearwallsto reachacceptable values of normal-izedwalldistancesy+,butalsoimprovespressurelosspredictions acrosstheswirler,whichisofparticularimportanceforthe deter-minationofthephasebetweenacousticandvorticalperturbations,

φ

uˆ_{θ −ˆ}v.Asaresult,a2%errorontheswirlerpressurelosswithM3

isobtainedifcomparedtoexperiments(seeTable2).

Intermsofmeshadaptation,Table1,onecanobservethat the transitionfromM1 toM2 reducedthe characteristiclength scale

Table 2

Mean pressure loss obtained with meshes M1, M2, M3 and in experiments. For numerical simulations, the mean pressure loss is calculated as the differ- ence between the inlet and outlet surface averaged pressures.

Case Pswirler (Pa) difference with Exp. (%)

M1 465 39

M2 364 9

M3 341 2

Exp. 335 NA

within the swirler, while the transition from M2 to M3 mainly impacted the injector and flame regions. To further validate M3, a fourthmeshM4 wasalso createdto assessmesh invariancein reactive simulations. Forthat purpose,mesh characteristic length scalesintheswirler,injectorandflameregionshavebeendivided bytwocomparedtoM1,yieldinglowercharacteristiclengthscales intheseregions(seeTable1)butalsocomparablelocalrefinement comparedto M3.Asaresultanddespitean increaseinthe com-putingpowerneededtoachieve thesamephysicaltime,only mi-nordifferenceswereobservedbetweenM3andM4.Thereaderis referred tosupplemental material B forfurtherdetails. As a con-sequenceandunlesstoldso,allresultspresentedinthefollowing areobtainedfromsimulationsbasedonmeshM3.

4. FTFassessmentfromstationaryflamedata 4.1. Numericalsetupvalidationagainstexperimentaldata

Inthefollowing,LES resultsbasedonmeshM3 arefirst com-paredtoexperimentaldatainnonreactingconditionstoassessthe reliabilityofthenumericalsetupprior toanycombustion simula-tion. As alreadyshownin the last section, pressurelosses across the swirler are in very good agreement with experiments. LDV measurements available from experiments over a line located 3 mm abovethe chamber backplaneare usedto gauge the numer-ical prediction. It is worth noting that measurements were ob-tainedwithouttheenclosingchamberwalls,whilesimulationsare always fully enclosed. For comparison,LES data is averaged over 137 ms, i.e. approximately 8.6 inlet to injector backplane flow throughtimes.Dataarefurthermoreaveragedintheazimuthal di-rectionsincetheinjector diametertochamber widthratioislow, resultinginaquasiaxisymmetricflow.Results presentedinFig.3

showaverygoodagreementwithexperimental dataforallmean velocity components.Aslightoverestimationof theaxial velocity peaksispresent,aswellasminordiscrepancieswhenx/R>1for radialandazimuthal velocities,R₌10mmbeingtheinjector outer radius. RMS velocity components, Fig. 3(d)–(f) are also in good agreement inthe central region where thefirst series ofpeak is capturedbyLES.Again,discrepanciesarevisibleintheoutershear layer (x/R > 1) whereonly theRMS ofaxial velocity isobserved to properly match the experiment. The estimated swirl number fromLDV measurements is 0.8while the LES yields S=0.73.All theseminordifferencescanbeattributedtotheunconfined exper-imentalmeasurementsversusconfinedsimulations.Another possi-bleexplanationliesinthefactthatLDVmeasurementsareknown tointroduceasmallbiasforturbulentflows[55],andthe discrep-ancies observedhereare localizedin theturbulent regionthat is theoutershearlayer.Furthervalidationisavailablein supplemen-talmaterialC.Overall,LEScapturestheinvestigatedconfiguration swirlingflowfeatureswithsatisfactoryagreement,allowingto pro-ceedtothereactingconditionswithconfidence.

In its steady state reacting regime, LES as well as experi-mental observations indicate that the flame has a classical V shape and is stabilized few millimeters above the central bluff

Fig. 3. Comparison of velocity fields under cold flow conditions measured with LDV ( ) and obtained in LES ( ), 3 mm above the chamber back plane. (a), (b) and (c) : mean values, (d), (e) and (f) : RMS values of axial (left), radial (middle), and azimuthal (right) velocity components. Axial distance is normalized by the injector radius R = 10 mm.

Fig. 4. Non-reactive (left) and reactive (rights) axial velocity fields and velocity streamlines. Combustion strongly affects the flow and widens the inner recircula- tion zone.

body. The flow structure downstream the injector is of course drastically modified by the presence of the flame incomparison tocoldflowconditions.Theexpansionoftheburntgasesincreases theflowvelocityneartheflame,pushingtheflowouter recircula-tionzonesdownwardswhiletheinnerrecirculationzonebecomes bigger than in the non-reacting case. As a result, Fig. 4 shows that theflowangleattheinjector exitslightlyincreases.Figure5

presents the heat release ratedistribution on a vertical plane in themiddleofthechamber,obtainedfromLESfieldsaveragedover time(80ms)andintheazimuthaldirection,complementedbyan AbeltransformofOH∗signalsfromtheexperiment.

Fig. 5. Comparison of steady state flame shapes. On the left, LES heat release rate field averaged over 80 ms. On the right Abel transform of OH ∗_{signal recorded by a}

CCD camera with a narrowband filter centered around 310 nm, averaged over 100 samples.

Forthenumericalsimulation,thenormalizedheatreleaserate averaged inthe azimuthal direction andover 80ms is used, the normalizationvalue beingthemaximumvalue found inthe sim-ulation. Clearly, numerical andexperimental flame shapesare in goodagreement,thoughaminordifferenceinflameheightcanbe observed and may possibly be explained by the limited thermal dataavailable fortheLEStomatchtheexperiment.Indeed,in ex-perimentstheflameexhibitsalargerlift-off distance thaninLES, whichmayimplythatthebluff-bodytiptemperatureislowerthan thevalueestimatedfromsimulations.Themeanflameangleisalso wellreproducedbythesimulation.Secondarybranchesonthe up-perpartoftheflamefoundintheexperimentarereproduced nu-merically,showingthatthermalconditionsnecessarytotheflame stabilizationare correctlyreproduced by the LES [56,57]. Further comparisonsusingPIV dataona verticalplane3.5mm abovethe

Fig. 6. Schematic of the flame and associated quantities of interest : α, L f , H f , R f ,

R0 . The position of the ˙ Q center of mass ( x c , y c ) is shown.

chamberbackplaneareprovided assupplementalmaterialin an-nexD.Theyshow a verygood agreementatseverallocationsfor bothmeanaxialandazimuthalvelocities,indicatingthattheflow angleisindeedwellcapturedbythesimulation.

Overall, LESisabletocaptureboth coldandreactiveflow fea-tureswithgoodaccuracy,allowingtomoveonwithconfidenceon totheSFTFparametersestimationstep.

4.2.SFTFmodelparametersestimationfromstationarydata

LES results are used to probe geometrical quantities of inter-estneededtodeterminethereducedfrequency

ω

_∗=

ω

L2

f/

(

v

Hf

)

as wellasthehalfflameangle

α

.Awidevarietyofflamedimension definitions is available in the literature, the most common ones rely on isolevels of a variable representative of the flame front (typicallyheat release rateor a progress variable) inthe case of numericalsimulations.Inthispaper,dimensionsaredefinedfrom thelocationofthecenterofmassoftheheat releaserateQ˙ field obtainedfromtimeandazimuthallyaveragedsolutions[58].Given

N,the numberof nodesinthe solution,the centroidcoordinates (xc,yc) oftheheatreleaseratedistributioninaverticalplaneare givenby: xC= N k=1Q˙kxkyk N k=1ykQ˙k and yC= N k=1Q˙kykyk N k=1ykQ˙k (16) Coordinatesfortheflameanchoringpointare muchlesssensitive toits definition.It isheredefinedasthe lowestpoint inthe ax-ial directionwhere Q˙ isat least superior to 1% ofits maximum value. Figure 6 shows the position of the retrieved centroid of heatreleaseratedistributionandtheassociatedflamedimensions. Forthespecificconfigurationofthiswork,onegets:Rf/R0=1.0, Lf/R0=1.75,Hf/R0=1.44and

α

=34.8◦.Themeanaxialvelocity

attheinjector exitplane

v

=8_.78 m/sismeasured fromthe sta-tionaryunperturbedLEStocompletetheanalysis,yielding

ω

_∗/

ω

= 2.43ms.

The next criticalstepinthe SFTFconstruction istodetermine theaxial convectionvelocityUc−v of vorticalstructures alongthe

outershearlayeroftheswirlingjetexhaustingtheinjector.For

V-Fig. 7. Identification of the outer shear layer using the I 2 criterion for the reactive

case. Each white dot represent the local maximum at a given height y / R 0 .

shaped flames like the presentone, these structures are respon-sible for large surface area perturbations and thus, in the case ofa premixedflame,for themajor partofthe unsteadyheat re-lease[33,36].Onepossibilitytoassesstherealspeedofthese dis-turbances is touse a trackingalgorithm [59]. While theoretically appealing, this method has some limitations in a turbulent LES framework andrequires acoustically pulsed simulations. Another possibility arising for highly swirling flows is to use properties coming from solid mechanics theory.For solid bodies, the norm ofthesecond principalinvariant ofthedeviatoricstress tensoris used as a measure of shear. In the present case, swirl is strong enough (and possibly the injection tube narrow enough)so that the angular momentum flux prevails, resulting in a fully devel-opedturbulentpipeflowinsolidbodyrotation.Asimilarcriterion

[60] resemblingthe classical

λ

2 criterionfor vortexidentification

isthereforeused,thatisthesecondinvariant I2 ofthestrain rate

tensor_S,whichisinpracticecomputedas: I2=

1

2 SiiSj j− Si jSji (17)

Negative values of I2 indicate high shear regions. Applied to the

meansteadyreactingfield issuedbyLES,maximumnegative val-uesofI2 identifyacollectionofabscissastartingfromtheinjector

exituptothedistanceHfassociatedtotheheight ofthecentroid ofheatreleaseratedistribution.Thisyieldsacurveassignedasthe outershearlayertrajectoryshowninFig.7.Notethatother crite-riacouldbe usedtoidentifytheshearlayer,such asthenormof thestrain ratetensor,removing thepossiblehighswirllimitation ofthemethod.TheI2criterionwashowevershowntobe

particu-larly robustforhighswirlingflows. Alongthispath,thelocal ax-ial velocityatagiveny abscissaisretrieved andthereafternoted

v_l(y). The axial velocity component of vortical structures Uc−v is

thenevaluatedbyaveragingtheaxialvelocityalongtheshearlayer pathoverthedistanceHf,thatis:

Uc−v=_H1

f Hf

0

v

l

(

y

)

dy (18)

whereHf isthe flameheight (Fig.6). Using theavailable average LES fields yields Uc−v=6.46 m/s. The realpart of the correction

1.5

31r/2

1

+

Exp.

SFTFl 1

1 1.0

::0-

_�

�

1r:

=

'<ij

rn

_�

b.O

..c:

�

C.

�

_0.5

�

_n/2

�

�

0.0

0.0

50

100

150

200

250

50

100

150

200

250

f (Hz)

f (Hz)

Fig. 8. SFTFl mode! resulcs with parameters estimated from a single stationary unperrurbed LES : Œ,/Œ = 2.43 ms, a= 34.8°_{, K = 1.36, x = -0.33, � = 5.55 ms, q>o = 0} rad.

factor K for the FTF mode! therefore equals K = ii

/

Uc-

v

= 1.36. In

the absence of pulsed LES data, one can estimate the time delay

t'

by assuming that acoustic perturbations travel at the sound speed

c while azimuthal perturbations are convected at the local flow

speed

Uc

over a distance

8

=

50 mm between the swirler vanes

exit where acoustic/vorticity conversion occurs

(29)

and the injec­

tor exit plane ( see

Fig. 2

):

(19)

As a first approximation, the phase

ef>o

is simply nullified. By doing

so, the low frequency gain limit condition from

Eq. (10)

reduces to

X

=

-s

and the FrF phase is evidently forced to a null value in the

zero frequency limit. Finally, choosing

Uc =

ii as a first approxima­

tion yields

t'

= 5.55 ms. The choice of the convective velocity

Uc

is

still subject to discussions in the community. It has been observed

experimentally

(61)

and while trying to reproduce FrF from mod­

els

(26)

that the actual value may be 40-50% Iarger than the bulk

velocity in the injection device. Recently, Albayrak et al.

(62)

have

proposed an analytical expression for this quantity in the Iow ax­

ial wavenumber limit based on a modal decomposition of the lin­

earized Euler equations which resumes to:

Uc = ïï(1 + 2K/Ào)

(20)

where

K

is the circulation strength of the swirling flow and Ào

is the first eigenvalue of a characteristic Sturm-Liouville equation.

Applied to the current configuration, the above expression yields

Uc

= 1.SSii with

K

= 1145

ç

1 and Ào = 470, resulting in

t'

= 3.51

ms which does not comply with the time delay found in the LES

as will be presented in

Section 5

. For this reason, it was chosen

to use

Uc =

v.

No information is available for the determination of

the swirl fluctuation intensity parameter

X.

In the present paper

and as a first step, a value comparable to those found in

(28.41)

is

used : x

=

-0.33.

From the parameters estimated previously, one can obtain a

first estimation of the flame acoustic response, that is denoted as

SFrF1 and is shown in

Fig

.

8

. With only a single stationary flame

simulation, the mode( is able to depict the FTF gain and phase ten­

dencies over the frequency range of interest. In particular, correct

phase tendencies are already retrieved without the introduction

of unsteady perturbations, and values match the experiment for

f

2:-,

150 Hz. The frequency of the first local FrF gain minimum is

however not retrieved using SFTF1, which also shows in the phase

curve where the phase shift region is not well predicted. This point

is further detailed in the next section.

S. Forced simulations for more accurate SFTF predictions

5.1. Acoustically pulsed LES results

In this section, focus is made on obtaining the parameters ef>o,

X

and

f3

which require at Ieast one acoustically pulsed simula­

tion since they are related to dynamic features: the phase lag be­

tween acoustic and convective perturbations, swirl fluctuations in­

tensity, velocity disturbances decay. Building upon previous simu­

lations, and for verification purposes, pulsed LES have been per­

formed to assess the CFD capability to successfully capture the

flame response to flow perturbations. For these cases, the inlet

boundary condition is changed to impose a uniform velocity pul­

sation with a 30% RMS amplitude, corresponding to 23 m/s vari­

ations, with a unique frequency in the range 80-200 Hz. NSCBC

relaxation coefficients are set to Iow values to avoid any unwanted

acoustic reflection

(63)

. Flame dynamics during a forcing cycle is

well retrieved as indicated for f

=

180 Hz in

Fig. 9

, by comparing

azimuthally and phase averaged heat release rate fields with OH*

chemiluminescence data at various instants in the forcing cycle. In

particular, the flame angle and dimensions as well as the roll-up

motion at the flame tip are well reproduced.

The FrF as obtained from single frequency LES forcing is pre­

sented in

Fig. 10

and shows excellent agreement with reference

data from the experiments. The characteristic Iow and high FTF

gain regions of a swirled V-shaped flame anchored on a bluff-body

are well retrieved. It is therefore concluded that LES is able to cap­

ture pulsed reactive flow features with good accuracy.

5.2.

SFTF

mode/

parameters estimation /rom pulsed LES

To the exception of

x,

ail SFrF parameters can be roughly es­

timated from a stationary reactive simulation without acoustically

forcing the flow, giving a first estimation of the flame acoustic re­

sponse. It is shown in this section that performing a few or even

only one additional pulsed simulation to obtain further informa­

tion can improve the mode( accuracy. This method avoids numer­

ous single frequency forced simulations

(20,21)

or the need for

other identification techniques

(18,64)

. In addition, since quanti­

ties of interest only concern the injection system at the exit plane,

which may only be marginally affected by burnt gases expansion,

a single non reactive forced simulation may be sufficient for some

of these parameters.

As an alternative minimizing the number of assumptions

(

Eq

.

(

19)

)

, the phase

ef>a

_

0

between acoustic and convective per­

turbations can also be determined from a set of pulsed flow sim­

ulations. Such a study would of course defeat the purpose of

Fig. 9. Abel transform of phase averaged OH' chemiluminescence pictures from experiments (left parts) and phase averaged field of normalized heat release rate from LES (right parts) for a forcing frequency f = 180 Hz. Here the experimental results were obtained with a cane protruding 1 mm higher in the chamber, but should however still be comparable. The phase rp is defined from the velocity signal at the hot wire position : rp = 0 corresponds to null acoustic velocity while rp = TC /2 corresponds to a maximum.

1.5�---+

++e

�

o+

₊

o

+

+

+o

+

+Q'i�

�

s

0.0

+---�---50

100

150

200

250

f (Hz)

...::'.., C.

�

�

�

31r/2�---�

7r

n/2

50

+

++

<i+

+ ooei

+

0

100

150

f (Hz)

+ (b

+

O+

+

+

200

250

Fig. 10. Flame transfer function of the NoiseDyn bumer : gain (left) and phase (right). Single frequency pulsed LES data (grey circles) is compared to experimenral measure­ ments (black crosses). Bulk velocicy T1 = 5.44 m/s, (fl/Tl)"-"5 = 30%.

using an analytical model which should avoid running several

costly simulations, but the study is performed as another valida­

tion test here. The phase between Fourier coefficients of axial and

azimuthal unsteady velocity signais integrated on the injector exit

plane is used for a set of eight forcing frequencies, giving the lin­

ear regression presented in Fig. 11. The time delay -r

=

5.54 ms and

phase at the low frequency limit

</>o

= 0.03 are found to be very

close to those obtained using the simple one-dimensional propa­

gation model from Eq. (19) leading to -r

=

5.55 ms [29). This result

may however not be satisfactory since the first local gain mini­

mum lies around

f

= 120 Hz and the maximum gain deviation is

attained when swirl fluctuations are maximal, that is for

</>a₀

-v

=

:,r

[65). Using Eq. (19) shows that a value close to -r

=

1/(2/)

=

4.17

ms would be expected if

</>o =

O. ln [28),

<l>û -ÎI

was evaluated ex­

perimentally at the base of the flame for difÏerent bulk velocities,

reporting values of -1 and -1.5 rad for

</>o

with forcing frequen­

cies going as low as 30 Hz. lt is then important to notice that ve­

locity profiles plotted in

Fig. 12 are not fiat at the injector outlet,

and that considering only bulk quantities may not be satisfactory.

lndeed, swirl fluctuations should be considered where they pre­

ponderantly affect the flame. ln the particular case of the Noise­

Oyn confined swirled V-flame, this region is the edge of the injec­

tor wall where large vortical structures are created and travel along

the shear layer and perturb the flame surface. Accordingly, one can

see

</>o

as a phase lag between the bulk oscillation signais and sig­

nais obtained at a particular radial location close to the wall, here

chosen as a point 0.5 mm away from the injector wall on the in­

jector exit plane, see Fig. 12. lt can be evaluated as:

</>o = <l>as-Îi(yinj,

0.95Ro) - 5

1

[

<l>as-Îi(yinj, r)dSe

e

1s

,

= <l>as-Îi(yinj, 0.95Ro) -

w-r

(21)

where

Se

designates the cross section area at the burner outlet

_Yinj·

A maximum deviation of less than 10% in

</>o

is obtained when

the probe position changes bY 0.3 mm. Larger deviations are seen

when using locations doser to the wall, depending on the local

•> 1

271'

,if 7r

50

100

150

f (Hz]

0

_LES

linear lit

200

250

Fig. 11. Phase becween axial acoustic (v') and azimuthal convective (uô) velocity perrurbaàons ac the injector exit plane from surface averaged data for different pulsed LES frequencies.

Table 3

Phase q,0 as obtained from LES data for different frequen­

cies using Eq. (21 ).

![Hz) 100 120 150 180

4>o [rad) -0.832 -0.905 -0.887 -0.874

mesh size. Using data from forced LES, one gets the results of

Table 3

, which tend to validate the assumption that

<Po

is almost

frequency independent in the studied pulsation range.

This means that this quantity can be obtained using only a sin­

gle pulsed simulation. The signais used to obtain these values are

shown in

Fig

.

13

, with

<Po

computed as

<Po

=

</>p

-

<Pb

using the fig­

ure notations. Inverting the equality

<f>a

₉_0

= rr with the newly ob­

tained values yields an evaluation of the frequency /

1

correspond­

ing to the FTF minimum gain:

1

<l>o

fi=---

_{2-r 2rr-r}

(22)

For

f =

120 Hz and

f =

180 Hz respectively,

Eq

.

(22)

yields /

1

=

115 Hz and /

1

= 116 Hz, which agrees well with the frequency

range 110 Hz � /

1 �

120 Hz obtained in the experiment,

Fig. 10

.

The last remaining parameter is the swirl intensity parameter

x.

Although its etfect on the mode! is quite straightforward, it is

still unclear how to directly measure it using either experiments

or LES. To remedy this situation, and since at least one acous­

tically forced reacting LES has to be performed to retrieve other

mode) parameters dealing with the system dynamics, it is pro­

posed to perform a pointwise optimization on both the FTF gain

(a)

_r

₌

_0.95Ro

1

1

1

1

1

1

1

1

YinJ--

=-=i

(b)

15

vi'

10

5

>

0

and phase at a particular frequency to determine a suitable value

for

X.

Best agreement can only be achieved when using frequen­

cies corresponding to local extrema of the FTF gain. Since the first

local gain minimum frequency is known from the previous step,

one can perform LES at this particular frequency and use it as a

target One advantage for using the lowest frequency known ex­

tremum lies in the fact that at this stage, no high frequency acous­

tic decay is taken into account and thus, the obtained

X

value is

more likely to be a good estimate. Once again, it is made use of

the data of the pulsed LES at

f

= 120 Hz in conjunction with an

optimization algorithm to determine the best value for the SFTF

mode! to match the FTF gain and phase as obtained from LES at

this particular frequency. The value

X

= -0.368 is obtained as the

optimal one.

With the addition of

<Po

and the optimization process on

x,

the

SFTF mode! reaches a higher level of complexity, here denoted as

SFTF2 and shown in

Fig

.

14

.

5.3.

SFTF mode/ with a spatial decay for velodty disturbances

Finally, the spatial decay rate of the velocity perturbation am­

plitude,

{3,

can be evaluated by means of a single pulsed cold flow

simulation with the same 30% RMS amplitude as in the reactive

case presented in the previous section. The FTF mode! derivation

was done assuming a clear separation between fresh and hot gases,

which is obviously not the case for a confined swirl burner where

outer recirculation zones contain hot gases. For this reason cold

flow simulations were preferred to determine

{3.

ln this case, two

forcing frequencies are investigated : /

=

120 Hz and

f

=

180 Hz

corresponding to the identified local minimum and maximum am­

plitudes of the FTF gain (

Fig. 10

)

. Velocity disturbance amplitudes

are probed on a vertical line at

r/Ro

= 0.75 which corresponds to

the central line between the injector outer wall radius and the con­

ical bluff body top radius as shown in

Fig. 15

(a

)

.

Figure 15

(b

)

shows

that LES predicts a decrease of the amplitude as expected from ex­

periments.

Post-processing the LES data leads to

f3

=

0.184 for f

=

120 Hz

and 0.188 for

f = 180 Hz. These specific values were obtained by

fitting an exponential function of the form

Ae-YY

so that

f3

=

Y

u

tv

where Uc

-vlco1d

= 3.83 m/s cornes from the technique described in

Section 4

for the unperturbed cold LES fields. Embedding the de­

cay rate in the SFTF mode! should allow for better gain predic­

tion at relatively high frequencies. ln the following it is chosen to

proceed with the value obtained for f = 120 Hz as this particular

frequency was already studied for reacting conditions. The optimal

value for

X

is of course not the same with the spatial decay

corn-0.6

0.8

r/Ro

1.0

0

_r,o=O

0

_{r,o=27r /5}

0

r,o=41r/5

0

_{r,o=61r /5}

•

r,o=81r /5

Fig. 12. (a) Sketch of the burner with the position of the integration surface height marked as Ymi and the probe near the outer wall represented

by

a black cross, located 0.5 mm

away

from the wall. (b) Axial velocity profile at axial position Yini for different phases of the forcing cycle (q.,

=

0 corresponds to a maximum velocity at the hot wire posiàon): àme evolving non homogeneous profiles indicate chat there may be a phase shift becween the bulk signal oscillation and the oscillaàon ac r

=

0.95R0•

ûxfü;

(bulk)

ûe

/'üë

(bulk)

--•--

-ûx/u,.,

(probe)

--•--

ûe/'üë

(probe)

(a)

0.6

b

0.6

0.4

0.4

0.2

0.2

1

l::i

0.0

0.0

<::Ï

_-0.2

_-0.2

-0.4

-0.4

-0.6

₁

₂

₃

-0.6

₀

₂

₃

t/T [-]

t/T [-]

Fig. 13. Normalized axial and azimuthal velociry signais on the injector exit plane (bulk, solid line) and on a probe 0.5 mm away from the outer injecror wall (dashed line with markers) for (a) f = 120 Hz and (b) f = 180 Hz. These signais correspond to ,J,o = -0.905 rad and

4>o

= -0.875 rad respectively, with ,J,o = q,9 -4>,,.

I

LO i:

-�

!::

0.5 50 100 150 f (Hz)

-g

..!:,.

�

r..

200 250 3,r/2

1

+

Exp. - SFTF2J

"

,r/2 0.0 50 100 150 200 250 f (Hz)

Fig. 14. SITT2 mode! results with parameters estimated from a stationary LES and an additional reactive forced LES at f = 120 Hz : w,/w = 2.43 ms, a= 34.8°_{, K = 1.36,} x = -0.368, t = 5.55 ms, 4>o = -0.905 rad.

(a)

_r

₌

_{_}

_0.75Ro

1 1 1 1 1 1

Yinj--

,•

,. 1" 1 1 1 1

(b)

6

5

- LES, f=120 Hz

_{- LES, f=180 Hz}

4

.B..

3

<>

-2

1

0

_o.o

0.5

1.0

1.5

2.0

(Y-Yiuj}/Ro [-]

2.5

3.0

Fig. 15. (a) Schematic of the central line located at r/Ro = 0.75 used for the amplitude decay evaluation. (b) Acoustic velociry amplitudes on the same line, starting from the injecror exit plane. Dotted Unes show the best fit for each frequency in the form Ae-YY.

ponent addition. The method is therefore iterated once gain, yield­

ing

X

= -0.336. This new methodology requiring a non-reactive

pulsed simulation for the decay rate determination is here denoted

as SffF3, and results are shown in Fig. 16. By stepping up to SFTF3,

the FTF gain for frequencies

f �

150 Hz matches reference data

due to the fact that the spatial/high frequency velocity perturba­

tion decay is taken into account. The phase curve in

Fig. 16

is

on the other hand only marginally modified compared to the re­

sults from SFTF2 shown in

Fig. 14. From these observations, it is

concluded that the SffF strategy constitutes a modular analytical

ffF mode( for swirled V-flames featuring different accuracy levels

depending on the number of simulations the user can afford. ln

any case, it remains less computationally intensive than perform­

ing several single frequency forced simulations as usually needed

to reconstruct the whole fff. Compared to a reconstruction based

on system identification techniques

[18,64)

, the method developed

in this work is not restricted to vanishingly small perturbation lev­

els and can be used to determine the frequency response of pre­

mixed swirled flames submitted to flow rate modulations of any

finite arbitrary amplitude. It is recalled that the results were ob­

tained here for a fixed perturbation RMS level tl

/ÏI

=

0.30.

Ali input parameters for the SFTF mode( determined using

methodologies SffFl. SFTF2 and SffF3 are summarized in

Table 4

for the sake of completeness, while a brief summary of each

methodology ability to represent the flame response is available

in

Table S

.

From these results, and depending on the available computa­

tional resources, one can choose whether a single simulation is

31r/2r:::=======:::::::----7

1 + Exp. -SFTF3 I 1 1.0 _...!:!.. 1r

.!3

_.,

<>

�

on ..c:

"'

i:l,

f;;

0.5

_�

_"'

rr/2 0 .0 0.0 50 100 150 200 250 50 100 150 200 250

f

(Hz)

f

(Hz)

Fig. 16. SFTF3 mode! results with parameters estimated from a stationary LES (SFfF2), and two additional forced LES at f

=

120 Hz for bath reactive and non reacrive conditions: w./w

=

2.43 ms, a= 34.8°_{, K}

=

_{1.36 +0.25i, x}

=

_{-0.336, t}

=

_{5.55 ms,}

<Po=

_{-0.905 rad.}

Table 4

SFTF parameters as determined from (a) a single reacring srationary LES or (b) us­ ing pointwise optimization on an additional reactive pulsed LES at f

= 120 Hz and

(c) with two additional cold/reactive pulsed LES at f

= 120 Hz.

Case w,Jw [msj a (degj K _X t [msj <Po [radj

(a) SFTFt 2.43 34.8 1.36 -0.33 5.55 0.0

(b) SFTF2 2.43 34.8 1.36 -0.368 5.55 -0.905 (c) SFTF3 2.43 34.8 1.36 + 0.250i -0.336 5.55 -0.905 Table 5

Overview of the FTF reproduction accuracy using an increasingly complex evalua­ tion of SFTF mode] parameters.

Case (a) SFTFt (b) SFTF2 (c) SFTF3 nb. of LES 3 4 Agreement on gain moderate moderate good Agreement on phase moderate good good

sufficient (Sfffl ), or if accuracy is sought, if it is preferable to run

additional forced simulations to obtain a better representation of

the ffF gain and phase (Sfff2 and Sfff3). Assessing the spatial de­

cay rates of velocity disturbances, SFrF3, yields a complete descrip­

tion of the flame acoustic response at the cost of an additional non

reactive pulsed simulation at any frequency compared to SFrF3.

5.4. Effect of the perturbation amplitude

Ali analyses performed up to this point have been made for a

fixed RMS perturbation level

vjii

= 30%. Jt is well known that while

the flame acoustic response remains unchanged for low amplitude

perturbations, for greater amplitudes, nonlinear interactions mod­

ify the FrF gain and phase

[6,15,17,20,22]. In the particular case of

fully premixed swirled flames, studies

[15.48]

show that increas­

ing the forcing amplitude barely affects the position of the gain

extrema of the FrF, but regularly reduces the gain for frequencies

higher than the one associated to the first local gain minimum.

The FrF phase lag remains unaltered by the forcing level. When

it cornes to the SFrF model, this implies that the time delay r.

but also the phase

<j,0

controlling the position of this local mini­

mum do not change when increasing the forcing level. The two re­

maining parameters which can therefore potentially account for ef­

fects of non-linearities are

f3

and

x.

Note also that the SFrF model

itself was derived from linear acoustics theory and should there­

fore not be suitable for high forcing levels. Despite this theoretical

limit, the model parameters are here determined from LES, which

is intrisically a nonlinear flow solver. Jt is therefore interesting to

gauge the mode! ability to handle different forcing amplitudes. ln

order to validate the aforementioned assumptions, two additional

non reacting pulsed simulations are run with RMS forcing levels

10.0

en

7.5

<>

5.0

2.5

0.00

-e-

u: /u

=

_10%

-e-

V:

/u

=30%

-e-

V:

/u

=55%

1

2

(y

-

Yinj)/�

[-]

3

Fig. 17. Acoustic velociry amplitude at r/Ro

=

0.75, starting from the injecter exit plane for RMS forcing levels 10%, 30% and 55% and forcing frequency f

=

180 Hz. Dotted lines show the best fit for each amplitude in the form Ae-YY.

Table 6

SFTF parameters as determined from SFTF3 procedure using f

=

180 Hz as a rarget for forcing levels 0/ü

=

10%, 30% and 55%.

Case w,Jw [msj a (degj K X t [msj ,Po [radj

0/11

=

10% 2.43 34.8 l.36+0.20i -0.341 5.55 -0.905

0/11 = 30% 2.43 34.8 l.36+0.26i -0.331 5.55 -0.905 0/11

=

50% 2.43 34.8 l.36+0.50i -0.346 5.55 -0.905

vjii

= 10% and

vjii

= 55%. For the latter, experiments have shown

a drop of the FrF gain from 1.05 for a 30% forcing level to 0.80

for a 55% forcing level at

f

= 180 Hz. The procedure described

in

Section 5.3

for the determination of the decay rate

f3

is ap­

plied once again for the two new forcing levels using phase aver­

aged data, yielding the decay curves shown in

Fig. 17

obtained for

f

= 180 Hz. This frequency was chosen as it corresponds to a local

FrF gain maximum. The corresponding decay rates are

f3

= 0.148

for 10%,

f3

= 0.188 for 30% and

f3

= 0.37 for 55% forcing amplitude,

indicating that higher dissipation occurs when increasing the per­

turbation strength. For each new set of parameters,

x

is again de­

duced from an optimization using LES obtained gain and phase val­

ues for

f

= 180 Hz. Newly obtained values for

X

do not differ sig­

nificantly when the forcing level is varied from 10% to 55% .. Note

also that for

Îl/ii

= 30%,

x

= -0.331 is obtained for an optimiza­

tion applied at

f

= 180 Hz, resulting in a coefficient very close to

x

= -0.336 obtained for

f

= 120 Hz in

Section 5.2

. Final parame­

ters for the three considered cases are summarized in

Table 6

.

Resulting FrF gain and phase curves for 10% and 55% forcing

levels are shown in

Fig. 18

.