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Numéro d’ordre : 2458THÈSEprésentée pour obtenir le titre deDOCTEUR DEL’INSTITUT NATIONAL POLYTECHNIQUEDE TOULOUSEEcole doctorale:Spécialité:Directeur de thèse:TYFEPDynamique des FluidesThierry POINSOTPar M. Alexis GIAUQUEFONCTIONS DE TRANSFERT DE FLAMMEET ENERGIES DES PERTURBATIONSDANS LES ECOULEMENTS REACTIFSSoutenue le 14 Mars 2007 devant le jury composé de:G. SEARBY Directeur de recherche à l’IRPHE de Marseille RapporteurC. BAILLY Professeur à l’Ecole Centrale de Lyon RapporteurF. VUILLOT Ingénieur/Chercheur à l’ONERA/DSNA ExaminateurS. DUCRUIX Professeur à l’École Centrale de Paris ExaminateurF. NICOUD Professeur à l’Université de Montpellier II ExaminateurT. POINSOT Directeur de Recherche à l’IMF de Toulouse Directeur de ThèseRéf. CERFACS : TH/CFD/07/15

Figure 1 - Photography of a ”real” wood fireFigure 2 - Picture of a ”simulated” methane flame (detail of figure 4.11)

RésuméCes travaux de thèse présentent une étude numérique (utilisant la Simulation aux Grandes Echelles)des instabilités de combustion. L’objectif général est d’en approfondir la compréhension en développantde nouveaux concepts physiques ainsi que de nouvelles méthodes d’analyses. Deux aspects principauxdes instabilités de combustion sont traités :• Les Fonctions de Transfert de Flamme (FTF)Quatre méthodes différentes pour la détermination des Fonctions de Transfert de Flamme (FTF)sont évaluées à l’aide de la Simulation aux Grandes Echelles (SGE). La méthode HF-FFT, reposantsur un forçage harmonique de la chambre de combustion et sur l’analyse de Fourier prévoitdes délais de combustion globaux en accord avec l’expérience. Cette méthode fournit aussi la FTFlocale (c’est à dire la réponse de chaque point de la chambre à une excitation en entrée) et permetde localiser les zones ayant la plus forte réponse à l’excitation. Elle permet aussi de connaître larépartition spatiale des délais de cette réponse. Cette méthode devrait donc se révéler utile pourl’analyse de configurations dans lesquelles la réponse de la flamme n’est pas compacte par rapportà la longueur d’onde de l’excitation.Une nouvelle méthode WN-WH, reposant sur le forçage par un bruit blanc filtré de la chambrede combustion et une analyse utilisant la relation de Wiener-Hopf est comparée avec succès à laméthode HF-FFT. Bien que WN-WH semble plus difficile à mettre en oeuvre que HF-FFT, sonavantage principal réside en ce qu’elle donne accès au spectre de fréquence de la FTF locale sansaugmenter le coût de calcul.Cette étude est en lien étroit avec l’analyse de stabilité des chambres de combustion. En effet,la FTF a une influence sur la fréquence et les taux d’amplification des modes dans les outilsnumériques utilisées pour déterminer la stabilité des brûleurs. Cette étude montre comment construirecette FTF qui est un élément essentiel de ce type d’analyse.• Les énergies des fluctuations et les critères de stabilité dans les écoulements réactifsPoursuivant les travaux de Chu [23] et de Myers [88], cette thèse développe une nouvelle équationde conservation pour une énergie nonlinéaire des fluctuations en combustion. Un nouvel outilde post-traitement modulaire est utilisé pour vérifier la fermeture des équations des énergies desfluctuations sur des flammes laminaires mono et bidimensionnelles .Cet outil donne par ailleurs accès à tous les termes physiques et numériques responsables del’évolution de ces énergies dans l’écoulement. Pour chaque équation, les principaux termes”source” sont identifiés et cette analyse fournit deux critères de stabilité pour les écoulementsréactifs. Ces critères sont validés sur le cas d’une instabilité se développant dans une chambre decombustion bi-dimensionelle.Le premier critère étend le critère de Rayleigh linéaire en y introduisant l’influence de la fluctuationdu flux de chaleur. Ce travail fournit donc un outil d’étude linéaire de la stabilité deschambres de combustion.Il montre également que l’équation de conservation de l’énergie entropique des fluctuations nepeut pas être linéarisée dans les cas réactifs en raison de l’amplitude des fluctuations locales detempérature. Le deuxième critère est donc non linéaire afin d’inclure l’influence de l’énergieentropique sur la stabilité de la flamme. Ce critère permet d’obtenir des informations sur la

stabilité de la chambre de combustion lorsqu’aucune linéarisation de l’écoulement n’est possible.Mots clefs : Simulation aux Grandes Echelles (SGE), Instabilités thermo-acoustiques, Fonctionsde Transfert de Flamme (FTF), Energies des fluctuations dans les écoulements réactifs, Critères destabilité en combustionAbstractThe general objective of this thesis is to extend the understanding of combustion instabilities by testingmodels, physical concepts and numerical procedures, and by providing new numerical post-processingtools to do so. Two main aspects of combustion instabilities are studied and lead to many outputs.• Flame Transfer Function (FTF)Four different methods for the determination of Flame Transfer Functions (FTFs) in LES havebeen tested. HF-FFT method, based on harmonic flame forcing and FFT post-processing givesresults of global combustion delays that compare well with the experiments. This method alsoprovides the local FTF which gives valuable information about the amplitudes, delays and maximumlocations of the local flame response. This method should reveal useful for configurationswhere the response of the flame is not compact compared to the characteristic wavelength of theexcitation, as for distributed reacting cases.A new WN-WH method based on filtered-white noise forcing and post-processing using theWiener-Hopf relation is successfully compared to HF-FFT. Though this method should be handledwith care, its main advantage is that it gives access to the frequency spectrum of the local FTFwith no additional computational cost.An important aspect of this study is its link with stability analysis of combustors. Obviously, FTFsdo have an influence on the frequency and amplification rates of modes in the numerical methodsused for combustor stability. This study shows how to construct FTFs which are an important”brick” of acoustic analysis.• Disturbance Energies and Stability Criteria in reacting Flows.Following the works of Chu [23] and Myers [88] for non-reacting flows, a new nonlinear conservationequation for a disturbance energy in gaseous reacting flows is derived.A new modular post-processing tool is used here to check the balance closure of disturbance energieson laminar 1D and 2D flames. This tool gives access to all the physical and numerical termsresponsible for the evolution of disturbance energies in the flow.For each equation, major terms are identified and this work proposes two stability criteria for reactingflows. These criteria are validated on the case of an instability developing in a 2D reacting4

configuration.The first criterion extends the linear Rayleigh criterion by taking into account the influence of thefluctuation of the heat flux. This work therefore gives a relevant linear tool for the study of combustionchambers stability.Besides, it also shows that the entropy disturbance energy cannot be linearized in reacting flowsbecause of the local amplitude of temperature fluctuations. The second criterion is therefore nonlinearto include the influence of the entropy disturbance energy on the global stability. Thiscriterion gives relevant information on the stability when no linearization of the flow is possible.Keywords : Large-Eddy Simulation (LES), Thermo-acoustic instabilities, Flame Transfer Function(FTF), Disturbance Energies in reacting flows, Stability Criteria in combustion5

ContentsRemerciements 11List of symbols 13Introduction 17Objectives & Organisation 28I Description of the numerical tool 331 Description of the numerical tool AVBP 351.1 Governing Equations and hypothesis for DNS . . . . . . . . . . . . . . . . . . . . . . . 351.2 Governing equations for LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.3 The Thickened Flame (TF) model for LES . . . . . . . . . . . . . . . . . . . . . . . . 501.4 General aspects of the Boundary Conditions in AVBP . . . . . . . . . . . . . . . . . . . 551.5 Artificial Viscosity in AVBP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.6 Cell-Vertex Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60II Flame Transfer Functions 632 Linear models for FTFs (Flame Transfer Functions) 692.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

CONTENTS2.2 Models for FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Measurement methods for FTF in LES 813.1 LES of FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.2 Postprocessing methods for local FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3 Preliminary comparisons of postprocessing methods. . . . . . . . . . . . . . . . . . . . 854 Transfer functions of flames 954.1 Configuration A : laminar planar premixed flame (1D) . . . . . . . . . . . . . . . . . . 974.2 Configuration B : laminar V flame (2D axi-symmetric) . . . . . . . . . . . . . . . . . . 1014.3 Configuration C : turbulent burner in cylindrical chamber (3D) . . . . . . . . . . . . . . 1044.4 Configuration D : turbulent burner in a 15 ◦ sector (3D) . . . . . . . . . . . . . . . . . . 1274.5 Evaluation of FTF measurements methods in LES. . . . . . . . . . . . . . . . . . . . . 138III Disturbance energies and stability criteria in reacting flows 1415 Introduction 1475.1 Previous studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.2 An advanced post-processing tool for LES : POSTTIT . . . . . . . . . . . . . . . . . . 1615.3 Examples of conservation equation balance closure . . . . . . . . . . . . . . . . . . . . 1716 Disturbance energies in flow 1756.1 Pressure-Velocity (PV) disturbance energy Eq.(1)[Eq.(6.10)] . . . . . . . . . . . . . . . . . . 1756.2 Entropy disturbance energy Eq.(2)[Eq.(6.18)] . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.3 Nonlinear disturbance energy Eq.(3)[Eq.(6.37)] . . . . . . . . . . . . . . . . . . . . . . . . . 1846.4 The choice of the baseline flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1896.5 Summary of disturbance energies conservation equations . . . . . . . . . . . . . . . . . 1917 Results 1958

CONTENTS7.1 Configuration A (1-D Reacting Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1997.2 Configuration B (2-D Reacting Case) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2107.3 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588 Stability criteria in reacting flows 2638.1 Evolution of disturbance energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2638.2 Linear criteria for stability : Rayleigh Criteria, Chu criterion . . . . . . . . . . . . . . . 2658.3 Deriving stability criteria from Eqs.(1), (2) and (3) . . . . . . . . . . . . . . . . . . . . 2688.4 Conclusion and prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272General Conclusion 275Bibliography 277Appendix 291A Linear conservation equation for Pressure-Velocity (PV) disturbance energy 291B Linear conservation equation for Entropy disturbance energy 297C Exact conservation equation for a nonlinear disturbance energy 301D Evaluation of numerical corrections for Eqs.(1), (2) and (3) 311E Short notations for balance closure analysis (Chapter 7). 313E.1 Equation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313E.2 Equation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314E.3 Equation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314F Publication in Proceedings of the Stanford CTR 2006 Summer Program. 3159

CONTENTSG Publication submitted to Combustion and Flame. 32910

RemerciementsJe remercie tout particulièrement Anne Clausse, sans qui cette thèse ne serait pas la même, ainsi quebeaucoup d’autres choses! Merci...Je tiens à remercier mon directeur de thèse, le Professeur Thierry Poinsot, pour m’avoir permisd’entreprendre cette étude à la fois théorique et numérique concernant les instabilités de combustion. Jele remercie tout particulièrement pour son aide dans l’étude des fonctions de transfert de flamme.Je tiens à remercier chaleureusement le Professeur Franck Nicoud pour son soutien et son aide, en particulierconcernant l’études des énergies des perturbations en combustion. Nous avons eu de nombreusesdiscussions fructueuses qui nous ont permis de mener à bien une étude intéressante de ces énergies.Je remercie Charles Martin pour son aide concernant l’étude de l’énergie acoustique en combustion. Il amis en place une première approche très utile de l’évolution de cette énergie. Le lecteur est renvoyé à sathèse dans laquelle figurent certaines analyses en lien avec les miennes.Je remercie également Laurent Benoit et Claude Sensiau pour leur aide concernant la compréhension dece qu’est un ”solveur fréquentiel des équations de Helmholtz”, chose qui n’était pas clair pour moi audébut de cette thèse....Je tiens à remercier Simon Mendez, Eléonore Riber, Valérie Auffray et Gabriel Staffelbach pour leursoutien, leur bonne humeur, les discussions parfois sérieuses et scientifiques et celles qui n’avaient rienà voir avec le sujet de ce mémoire... Merci pour tout ceci et un peu plus!Finalement, je tiens à remercier tous les doctorants, post-doctorants et personnels de recherche duCERFACS à la fois passé et présent pour l’ambiance chaleureuse, décontractée et professionnelle qu’ilsapportent avec eux chaque jour au laboratoire.

REMERCIEMENTS12

NomenclatureRoman charactersc vHeat capacity at constant volume∆H 0 jEnthalpy change of reaction jDDisturbance energy source term∆S 0 jEntropy change of reaction jD kSpecies diffusion coefficientK f ,jK r,jForward reaction rate of reaction jReverse reaction rate of reaction jE Efficiency function (Part I)E, e Total energyQ jRate progress of reaction je kSpecies energy⃗J, ⃗q k⃗m⃗q⃗u⃗V k⃗WA(n)A +A −Species Diffusive fluxMomentumHeat fluxVelocity vectorSpecies diffusion velocityDisturbance energy fluxLES code variable updateDownstream propagating acoustic waveUpstream propagating acoustic waveAl(n) LES code variable update linked with walllaw correctionse 1E de ske sFDisturbance energy in quadratic pressureand velocity fluctuationsGeneral disturbance energy in flowSpecies sensible energyDisturbance energy in quadratic entropyfluctuationsFlame Transfer FunctionF Thickening factor (Part I)F pF uFlame Transfer Function relative to pressurefluctuationsFlame Transfer Function relative to velocityfluctuationsB(n)cLES code variable update linked withboundary conditionsSound velocityF norm2 Flame Transfer Function (Normalization2)F norm Flame Transfer Function (Normalization 1)c pkSpecies heat capacity at constant pressureF resFlame Transfer Function for 1D analysisc pHeat capacity at constant pressuregFree enthalpyc vkSpecies heat capacity at constant volumeg sFree sensible enthalpy

LIST OF SYMBOLSg skSpecies free sensible enthalpyΩ TVolumic integrated heat releaseH, h Enthalpyω kSpecies source termh kSpecies enthalpyΦDissipation functionh skSpecies sensible enthalpyφEquivalence ratioh m skMolar sensible species mass enthalpyρMass densitynAmplitude of the Flame Transfer Functionρ kSpecies mass densityp PressureQ rrss ks m kTWW kX kY kRadiative source termGas constant of the mixtureEntropySpecies entropyMolar species entropyTemperatureMean molecular weightSpecies molecular weightSpecies molar fractionSpecies mass fractionGreek characters∆ Filter characteristic sizeδ Kronecker operatorδL0∆ xThickness of the premixed flameStandard mesh size˙ω T /ω T Heat releaseγ Specific heat ratioλ Heat conduction coefficientστViscous tensorTime delay of the Flame Transfer Function(Part II)τ, ¯τ Stress tensorΞWrinkling factorDimensionless numbersRL eMP rP t rR eRe tSS c,kS t c,kOperators¯f tDfDtUniversal gas constantLewis numberMach numberPrandtl numberTurbulent Prandtl numberReynolds numberTurbulent Reynold numberSwirl numberSchmidt numberTurbulent Schmidt numberTurbulent component of the Reynolds filteredvariable fTotal derivative of scalar fµ Molecular viscosityˆfFourier transform of variable fµ t Turbulent molecular viscosityf Reynolds filtering operator (Part I)14

List of symbolsfTime average of variable fNSCBC Navier-Stokes Characteristic B.C.∂Partial differentiation operatorPDFProbability Density Function⃗∇. ⃗ fDivergence of vector ⃗ fRANS Reynolds Averaged Navier-Stokes⃗∇fGradient of scalar fSGSSub-Grid Scale˜fFavre filtering operatorTFThickened FlameSubscriptsTFLES Thickened Flame model for LESτ ijf 0f 1u iu jij component of the stress tensorZeroth order of the linear decomposition ofvariable fFirst order of the linear decomposition ofvariable fComponent i of vector ⃗uComponent j of vector ⃗uTTGC Two-step Taylor-Galerkin ColinWALE Wall Adapting Linear Eddy (model)WN-FFT White Noise (forced) - Fast FourierTransform (post-processed)WN-WH White Noise (forced) - Wiener-Hopf(post-processed)Superscripts¯fTensor notation of ff ′ Subgrid scale component (Part I)f ′Time fluctuation of variable fShort namesAVIBCCFDDNSDTFArtificial VIscosityBoundary ConditionComputational Fluid DynamicsDirect Numerical SimulationDynamically Thickened FlameHF-FFT Harmonically Forced - Fast FourierTransform (post-processed)LESLWMPILarge Eddy SimulationLax-WendroffMessage Passing Interface15

LIST OF SYMBOLS16

IntroductionBien qu’il apparaisse désormais évident que les réserves de combustibles fossiles s’épuisent, leurutilisation pour la production d’électricité et pour le transport reste largement majoritaire par rapportaux formes ”renouvelables” d’énergie. En effet, l’immense majorité des systèmes de production despays développés ou en développement reposent actuellement sur la combustion et la transformation desdérivés organiques fossiles. Conscients des dangers de la dépendance énergétique qui en résulte, denombreux chantiers de recherche se sont ouverts pour tenter de s’en affranchir.Concernant la production d’électricité de masse, le nucléaire, malgré les risques qui l’accompagnent 1apparaît comme le seul candidat capable de soutenir le niveau de consommation actuel. Les récentesdéclarations de l’exécutif américain 2 ainsi que celles de la Commission Européenne montrent le regaind’intérêt pour ce mode de production d’énergie. Le lancement récent du projet ITER (reposant surla domestication de la fusion thermonucléaire) fournit un espoir supplémentaire de pouvoir disposerun jour (le projet doit fournir des résultats industriels à l’horizon 2040-2050) d’une source d’énergiequasi-inépuisable et dont les matières premières seraient mieux distribuées.Cependant, du fait de l’organisation économique actuelle, et malgré l’augmentation inévitable deson coût, l’utilisation d’énergies fossiles reste prépondérante dans bien des domaines. Parmi leursnombreuses utilisations, ces énergies servent deux intérêts majeurs :• Leur combustion entre en jeu dans la production d’électricité par les centrales thermiques.• Elle fournit également un travail mécanique utile servant aux transports, qu’ils soient terrestres ouaériens.Récemment, la production d’électricité par turbine à gaz a connu un regain d’intérêt en lien avec ladérégulation du marché de l’énergie. En effet, grâce à l’augmentation de son rendement (cf Fig.3) et àsa grande flexibilité (temps de livraison, temps de maintenance, etc...) ce mode de production d’énergieest devenu extrêmement concurrentiel pour les ”petits producteurs” d’électricité.1 les principaux risques sont liés aux accidents nucléaires en production (Tchernobil (URSS), Three Mile Island (USA),Saint-Laurent (France), Tokaimura (Japon)....) et aux risques de contamination après utilisation du combustible, ces matériauxnécessitant un stockage pouvant durer plusieurs dizaines de milliers d’années.2 Les Etats-Unis ont pourtant connu un accident nucléaire majeur en 1979 (Three Mile Island).

INTRODUCTIONFigure 3 - Evolution des coûts de production en fonction de la quantité d’électricité produite à l’aide des turbinesà gaz. Chaque point de la courbe représente le meilleur compromis à une date donnée. Depuis le début desannées 1990, il est devenu plus rentable de produire de l’électricité à l’aide de turbines de faible puissance.Source: ”Charles E.Bayless, Less Is More: Why Gas Turbines Will Transform Electric utilities. Public UtilitiesFortnightly 1994”Bien que des énergies alternatives existent concernant le transport routier (systèmes hybrides essenceélectricité)et que des systèmes innovants voient le jour (pile à combustible, hydrogène, solaire, etc...),la combustion d’essence, de diesel ou de fuel n’a pas encore de concurrent économiquement sérieux.Quels sont alors les points à améliorer pour nous permettre de vivre au mieux avec ce modede production et de consommation d’énergie et d’en limiter les méfaits sur l’environnement?Une solution évidente est la limitation de la consommation de carburant de tous les types de chambresde combustion en améliorant leur rendement, permettant ainsi d’utiliser au mieux les réserves présenteset de diminuer les émissions de dioxyde de carbone (CO 2 ) (produit inévitable de la combustion decomposés carbonés et gaz à effet de serre). Cependant, le CO 2 n’est pas le seul polluant produit parla combustion d’hydrocarbures et de nombreuses études s’attachent aussi à limiter la production d’uneautre famille de polluants : les oxydes d’azote (NO x ).Ces dérivés de la combustion d’hydrocarbures dans un milieu riche en azote (comme c’est le cas dansl’air) sont à l’origine d’un cycle de réactions chimiques qui mènent à la diminution de la concentrationen ozone(O 3 ) dans les couches élevées de l’atmosphère. Or l’ozone empêche une partie du rayonnementultra-violet de se frayer un chemin jusqu’à la surface de la Terre. Lorsque la concentration en O 3des hautes couches de l’atmosphère diminue, une plus grande partie ce rayonnement nous parvient etentraîne une augmentation de l’incidence des cancers de la peau. Paradoxalement, au cours de ce mêmecycle chimique, de l’ozone peut être créé au niveau du sol menant ainsi à la formation de brouillardsd’ozone. Or l’ozone est toxique par inhalation, et ces brouillards d’ozone entraînent cette fois desrisques respiratoires. La figure 4 résume ce cycle simplifié de l’ozone.Le monoxyde d’azote (NO) est l’un des représentants de la famille des NO x . La formation de NOdans les chambres de combustion a lieu à travers deux principaux mécanismes:18

IntroductionFigure 4 - Cycle simplifié de l’ozone incluant l’influence de NO originaires de la combustion. source: TheESPERE Associaton (Environmental Science Published for Everybody Round the Earth)• la formation ”rapide” de NO, due principalement à la réaction entre le radical OH ⋆ et la moléculeN2 dans les flammes pré-mélangées.• la formation ”thermique” du NO, dû à la réaction entre la molécule de O 2 et celle de N 2 .Ces deux mécanismes de formation encouragent à diminuer la température à laquelle a lieu la combustionpour limiter les émissions de NO. Cependant, aux basses températures se forme alors un autre polluant:le monoxyde de carbone (CO). Comme le montre la figure 5, il existe une température optimale decombustion à laquelle on parvient à minimiser à la fois la concentration en CO et en NO dans les gazbrûlés.Ces températures optimales de fonctionnement étant plus faibles que les températures stœchiométriques3 de combustion, les industriels se sont orientés progressivement vers des technologies decombustion à richesse 4 basse et pré-mélangées, ainsi qu’au développement de chambres de combustionétagées permettant une maîtrise locale de la richesse du mélange. Cependant, à des richesses faibles(la richesse moyenne de combustion est parfois plus faible que la limite d’extinction de la flamme), lacombustion est beaucoup plus sensible aux phénomènes d’instabilités. Dans ce cas, des études récentesont montré [126] que la quantité de polluants émis pouvait augmenter.Il convient alors de mieux comprendre les instabilités de combustion afin de permettre l’utilisationindustrielle d’une combustion plus pauvre et donc plus ”propre”.3 Température atteinte lorsque la combustion à lieu à une richesse unitaire4 La richesse d’un mélange caractérise le rapport entre la quantité de carburant et celle de comburant disponible. Elleaugmente avec la concentration de carburant.19

INTRODUCTIONIndices d’émissionCONormesEuropéennesPlage de températuresà faibles émissionsNOx1500 1600 1700 1800 1900 2000 2100 KTempérature de la zone primaireFigure 5 - Graphique montrant l’évolution des émissions de NO et CO en fonction de la température decombustion.20

IntroductionInstabilités de combustionL’étude de la minimisation des concentrations en polluants dans les gaz brûlés est un aspect majeurde la recherche actuelle en combustion. Cependant, celle-ci se heurte depuis de nombreusesannées au phénomène des instabilités de combustion. Comme on l’a vu précédemment, les brûleursactuels fonctionnant à des richesses globales faibles sont plus sensibles aux instabilités de combustion[27, 51, 76, 97, 109]. Dans ce cas, ils sont alors plus susceptibles de générer du bruit lié à lafluctuation périodique du dégagement de chaleur. Par ailleurs, lorsqu’une chambre de combustion estsoumise à une instabilité, le mélange est moins bon et il se crée alors des zones plus riches que lamoyenne et qui vont donc produire plus de NO [7, 36, 63, 126, 148]. La prévision des instabilitésde combustion et l’amélioration des modèles d’interaction entre la flamme et les divers phénomènesinstationnaires de l’écoulement sont donc actuellement deux points clés de la recherche en combustion.Sous le terme d’instabilités de combustion se cachent de nombreux phénomènes physiques qu’il convientde détailler ici. Les instabilités liées à la combustion peuvent être séparées en deux grandes famillessuivant qu’elles ne font intervenir que la flamme, ou bien qu’elles nécessitent un confinement de lazone de combustion pour prendre naissance. Les premières sont nommées instabilités intrinsèques et lessecondes, instabilités extrinsèques de flamme.Instabilités intrinsèquesDans la première catégorie sont recensés tous les types d’instabilités ne faisant intervenir que la flamme,sans considérer son éventuel confinement. Les instabilités intrinsèques de combustion ont comme origine,• soit un terme source de type ”Rayleigh-Taylor”, c’est à dire reposant sur la différence de densitéentre les gaz frais et brûlés,• soit un terme source lié à l’instabilité de ”Darrieus-Landau” ou ”Hydrodynamique” reposant surla déviation des lignes de courant en amont et en aval du front de flamme et menant à la courburedu front de flamme,• soit un terme source menant à l’instabilité de type ”Thermo-diffusive” reposant sur la différenceentre les coefficients de diffusion d’espèces et de chaleur.Clavin et Garcia [26] fournissent une analyse complète des instabilités intrinsèques dans les flammesplanes prémélangées. Ces travaux fournissent une relation de dispersion théorique qui montre que pourla plupart des vitesses de flamme laminaire, le taux de croissance de l’instabilité de Darrieus-Landau estpositif sur une large bande de fréquence (figure 6). Clanet et al. en 1998 [25] ont effectué la premièreétude expérimentale de l’instabilité de Darrieus-Landau, et ont montré la validité du modèle théoriquede Clavin et Garcia pour des vitesses de flamme inférieures à 20cm/s et pour une longueur d’onded’instabilité fixée. La figure 7 montre un instantané du front de flamme perturbé par une instabilitéde Darrieus-Landau. Le phénomène est mono-fréquentiel comme en témoigne la périodicité spatiale des21

INTRODUCTIONfigures liées à l’instabilité. Il est à noter que l’instabilité de Darrieus-Landau est actuellement vue commel’un des mécanismes possibles de la transition déflagration-détonation 5 [72] en combustion, ce qui montreque ce type d’instabilité peut avoir une importante influence sur l’évolution du front de flamme mêmelorsque celle-ci est largement turbulente.Figure 6 - Taux de croissance réduit en fonction du nombre d’onde réduit de la perturbation. Source [25]Figure 7 - Front de flamme soumis à l’instabilité de Darrieus-Landau. Source [119]5 La transition déflagration-détonation se produit lorsqu’une onde de choc traverse un front de flamme se propageant à unevitesse faible. Il se produit alors une transition après laquelle le front de flamme suit l’onde de choc à une vitesse supersonique.22

IntroductionInstabilités extrinsèquesLa deuxième catégorie concerne les instabilités pour lesquelles le confinement est nécessaire. La manièreglobale dont ces instabilités se développent est résumé sur la figure 8. La flamme est l’élément centralde la boucle d’instabilité, amplifiant et transformant les différents types d’ondes présentes dansl’écoulement. Les instabilités en milieu confiné peuvent donc être ordonnées suivant le type d’ondeexcitée dans l’écoulement.A l’origine de l’instabilité, une faible fluctuation de l’écoulement amont vient perturber le front deflamme (figure 8-zone2). La flamme amplifie cette perturbation et crée principalement deux typesd’ondes se propageant dans l’écoulement (figure 8-zone3):• une onde acoustique se propageant à la vitesse u + c (u étant la vitesse de l’écoulement et c celledu son),• une onde entropique se propageant à la vitesse u de l’écoulement.Ces ondes rencontrent les murs et la sortie de la chambre de combustion (figure 8-zone4). L’ondeacoustique est en partie réfléchie, donnant naissance à une onde acoustique remontant l’écoulement àla vitesse u − c. L’onde entropique créée par la flamme peut aussi donner naissance à une onde acoustiqueremontant l’écoulement. 6 L’onde acoustique se propageant vers l’amont peut interagir directementavec la flamme et ainsi exciter une nouvelle onde entropique et/ou acoustique. Elle peut aussi remonterl’écoulement jusqu’au niveau de l’injection. Au niveau de l’injection, l’interaction entre l’acoustiqueet l’écoulement est complexe (figure 8-zone1). De nombreux travaux de recherche traitent ce problèmespécifique [132] et de manière générale, on observe une réflexion de l’énergie acoustique remontantl’écoulement sous forme d’une onde acoustique, d’une onde entropique ou d’une onde de vorticité (c’està dire une structure cohérente se propageant à sa vitesse de groupe α). Suivant les différents types defluctuations perturbant le front de flamme (figure 8-zone2), la boucle d’instabilité peut être fermée dedifférentes manières.• Il existe un transfert d’énergie possible entre les fluctuations d’entropie et les fluctuations acoustiques.C’est, par exemple, le cas d’une flamme subissant une fluctuation de richesse périodique.Celle-ci donne lieu à une fluctuation de dégagement de chaleur créant à la fois une onde acoustiqueet une onde entropique se propageant dans le sens de l’écoulement [140].• La flamme peut aussi transformer de l’énergie de vorticité en énergie acoustique. Ce phénomène alieu lorsque par exemple une structure cohérente plisse le front de flamme et crée une variation localedu dégagement de chaleur. Comme dans l’exemple précédent, cette fluctuation du dégagementde chaleur alimente l’énergie acoustique derrière le front de flamme [47].• Il a été observé une re-laminarisation dans les gaz brûlés pour certains écoulements d’injectionfaiblement turbulent. Ce comportement vient de l’expansion des gaz brûlés qui a pour effetd’écarter les lignes de courant et donc de diminuer le niveau de vorticité. Ceci explique pourquoi6 De manière générale, Kovasnay [64] montre que hors du domaine linéaire, il existe des échanges entre les énergies desperturbations liées aux fluctuations acoustiques, turbulentes et entropiques23

INTRODUCTIONla flamme tend à atténuer la vorticité qui la traverse. (flèches pointillées figure 8-zone3).Figure 8 - Schéma d’ensemble de la boucle d’instabilité de combustion dans les milieux confinés.Les mécanismes de fermeture de l’instabilité font intervenir différents types d’ondes se propageant àdes vitesses différentes. Le temps nécessaire à l’information pour parcourir le cycle d’instabilité n’estdonc pas le même suivant le type d’instabilité. Ceci est l’une des raisons pour lesquelles différentesfréquences d’instabilités sont observées dans les chambre de combustion. En effet, la fréquence del’instabilité ne peut être inférieure à la fréquence liée au temps de bouclage.La figure 9 présente une chambre de combustion simplifiée. La flamme est située à 10 cm de l’injectiondans une chambre de 50 cm de long. Le tableau 1 fournit pour cette configuration les vitesses et tempsde propagation des différents types d’ondes pouvant être impliquées dans le phénomène d’instabilité.Le tableau 2 résume (grâce à ces temps de propagation) les fréquences minimales possibles suivant letype d’interaction. Il est à noter que l’analyse ne comprend que les fluctuations colinéaires à l’axe dela chambre de combustion. On observe que les instabilités purement acoustiques sont principalementresponsables des fluctuations à plus haute-fréquence. Tandis que les fréquences intermédiaires (entre 100et 1000 Hz) sont liées à un mécanisme faisant intervenir un couplage entre l’acoustique et l’écoulement,via une transmission de la perturbation sous forme de fluctuation entropique ou sous forme d’unepropagation de vorticité (par exemple une structure cohérente). Dans son livre ”Combustion Theory”,Forman A. Williams [154] prend l’exemple des instabilités observées dans les chambres de combustioncryotechniques. Dans ces chambres, deux familles principales d’instabilités ont été observées :• le ”screaming” est un mode d’instabilité dont la fréquence est supérieure à 1000 Hz.• le ”chugging” dont la caractéristique est d’être basse fréquence (inférieure à 100 Hz), est principalementdû au couplage entre la chambre de combustion et le système d’injection.24

IntroductionCette différentiation en deux familles distinctes (”chugging” et ”screaming”) correspond bien à desmécanismes sous-jacents d’excitation différents comme on peut le voir dans le tableau 2. Ainsi,dans l’exemple choisi, une instabilité purement acoustique ne peut se développer pour des fréquencesinférieures à 675Hz. Pour expliquer la présence d’instabilités de plus basse fréquence, on doit considérerune interaction entre l’acoustique et l’écoulement. On observe (Tableau 2) que lorsque l’on invoque unestructure cohérente dans la première partie de la boucle d’instabilité, la fréquence minimale pouvant êtreexcitée diminue pour atteindre 191 Hz.Figure 9 - Chambre de combustion schématisée.zone type de d’onde vitesse de propagation (m.s-1) temps de propagation (ms)1 acoustique (amont) 400 0.251 acoustique remontante (amont) 300 0.331 entropique (amont) 50 21 vorticité (amont) 25-50 (vitesse de groupe de la structure) 2-42 acoustique (aval) 1000 0.42 entropique (aval) 100 42 acoustique remontante (aval) 800 0.5Table 1 - Temps caractéristiques nécessaires au différents types d’ondes pour parcourir les zones définies sur lafigure 9Outils d’analyse numérique des instabilités de combustionDeux principales méthodes numériques d’analyse peuvent être appliquées pour connaître la sensibilitéd’une chambre de combustion aux instabilités thermo-acoustiques : les codes d’instabilité et les critères25

INTRODUCTIONtype d’interaction temps total de la boucle d’instabilité (ms) fréquence minimale d’excitation (Hz)1A-2A-RA 1.48 6751A-2E-RA 5.08 1971E-2A-RA 3.23 3091E-2E-RA 6.83 1461V-2A-RA 3.23-5.08 191-3091V-2E-RA 6.83-8.83 113-146Table 2 - Fréquences minimales d’existence des instabilités suivant le type d’interaction. A:acoustique,E:entropique, V:vorticité, R:retour, 1,2:zones définies sur la figure 9d’instabilité.• Codes d’instabilitéLa première est quantitative et cherche à prévoir l’amplitude ainsi que les fréquences des instabilitéspouvant se développer dans la chambre de combustion. Il s’agit d’outils de résolutiondes équations régissant le comportement de l’écoulement perturbé. L’acoustique jouant un rôleprépondérant, il n’est pas surprenant que la plupart de ces outils ressemble à des codes d’acoustiquedans les domaines fermés où l’outil développé a pour but de donner les fréquences et les taux decroissance des modes instables. Au sein de ce type d’analyse, on peut distinguer trois grandesméthodes de résolution :– Les méthodes d’éléments en réseaux [8, 43, 44, 78, 108, 112, 145].Ces méthodes résolvent les amplitudes et fréquences des modes acoustiques longitudinauxdans chacun des éléments. Leur avantage est de permettre une étude paramétrique de la stabilitéprenant en compte des impédances complexes aux frontières du domaine. Cependant,cet outil requiert de connaître au préalable la forme des modes intervenant dans l’analyse etne convient pas lorsque la géométrie est complexe ou la fréquence des modes élevée.– Les méthodes directes spatio-temporelles - Simulation aux Grandes Echelles (SGE) [3, 19,28, 37, 57, 87, 102, 101, 138]La SGE fournit l’évolution spatio-temporelle des variables de l’écoulement. Elle prend doncen compte les interactions non linéaires pouvant se produire au sein du fluide. Cette méthodedemande une importante puissance de calcul et ne donne accès qu’au mode acoustique leplus amplifié. Par ailleurs, l’utilisation de conditions aux limites avec impédances complexespeut se révéler difficile.– Les méthodes de résolution dans l’espace des fréquences. [33, 68, 83, 90, 128, 129, 136]Ces méthodes fournissent les fréquences ainsi que les amplitudes complexes des modesacoustiques en linéarisant les équations de Navier-Stokes réactives. Le principal avantagede ces méthodes est de pouvoir prendre en compte des conditions aux limites faisant intervenirdes impédances complexes. Mais l’analyse de stabilité est linéaire dans ce cas et larésolution du problème dans le domaine non linéaire peut s’avérer complexe. Un élément26

Introductionessentiel de ces approches est le besoin de modèles décrivant la réponse de la flamme (c.à.d.le taux de réaction) à des perturbations acoustiques de la chambre. Cet élément appelé FTF(Fonction de transfert de flamme) est la difficulté principale dans ce type d’approche.• Critères d’instabilitéLe second type d’analyse est qualitatif. Il s’agit de déterminer si un critère simple, basé sur leniveau de fluctuation des variables primitives (pression, dégagement de chaleur, température, etc...)peut permettre de prévoir la présence d’une instabilité thermo-acoustique.Le premier critère de ce type a été développé par Rayleigh en 1878 [120] et prévoit qu’une instabilitéthermo-acoustique peut se développer si la fluctuation de pression est en phase avec celle dudégagement de chaleur. Depuis, de nombreux travaux se sont attaché à étendre ce critère qui ne permetpas toujours de détecter la présence de l’instabilité. Deux directions principales ont été suivies.La première consiste à étendre la notion d’énergie acoustique à des cas dans lesquels l’écoulementmoyen ne peut être négligé [16, 18, 85]. La seconde cherche à déterminer une équation de conservationnon linéaire pour l’énergie des fluctuations dans les écoulements [23, 88].Ces deux aspects, quantitatif et qualitatif, de l’étude des instabilités de combustion sont abordésdans cette thèse. Ces travaux s’inscrivent donc dans la continuité d’études précédentes [8, 82, 135,139, 143, 149] menées au CERFACS et utilisant l’outil numérique instationnaire (Simulation auxGrandes Echelles-Simulation Numérique Directe) comme :– un moyen de caractériser la Fonction de Transfert de Flamme (réponse de la flamme à uneperturbation de l’écoulement). Cette information permet d’accéder aux taux de croissancedes modes acoustiques en utilisant les méthodes de résolution dans l’espace des fréquencesprécédemment décrites.– un moyen de réaliser des expériences numériques mettant en évidence les principaux termessource des instabilités thermo-acoustiques et ainsi fournir un critère simple permettant dedétecter ces instabilités dans les écoulements réactifs.27

OBJECTIVES & ORGANISATION28

Objectives & OrganisationObjectivesThis work presents a numerical study (using both Large-Eddy Simulation (LES) and Direct NumericalSimulation (DNS)) of two main aspects of combustion instabilities. It is part of the undergoing studiesthat tend to address this issue for more than fifty years. The final goal of this long-lasting research effortis the understanding and the prediction of thermo-acoustic instabilities in combustion. The objectives ofthis thesis are:• the extension of existing and the development of new numerical post-processing tools for the studyof Flame Transfer Functions with the help of LES. As far as these transfer functions are concerned,only global or 1D responses are currently used in acoustic codes, assuming a compact reaction zonecompared to the acoustic wavelength. Yet, the flame is not always compact and distributed reactionzones should also be considered. There is therefore a need for a tool that could determine the localresponse of the flame to the incoming perturbations (i.e. the correlation between the inlet velocityfluctuation and the local heat release fluctuation). This tool would help to understand the influenceof the geometry and the injection system on the global response of the flame. This would alsobe a step towards the creation of an acoustic tool able to determine the linear stability of reactingsystems.• the derivation of new stability criteria in gaseous reacting flows. Recently, Nicoud and Poinsot [93]have pointed out the possibility that the classical Rayleigh criterion might not be relevant for allreacting flows since it does not take into account the disturbance energy created by the fluctuationof entropy. Following Chu [23], they derive a linear criterion based on the correlation betweentemperature and heat release fluctuations. The first objective here is therefore to test these linearcriteria on a 2D laminar unstable flame using Direct Numerical Simulations (DNS).Exact conservation equations for disturbance energies should also be derived and their balanceclosed to test the assumptions usually made when deriving stability criteria in reacting flows. Sincethe combustion phenomenon creates high temperature fluctuations, this work also aims at derivinga nonlinear stability criterion that could be used when no linearization of the flow is possible.

OBJECTIVES & ORGANISATIONOrganisationFirst, Part I provides a description of the numerical tool AVBP used during this work. Then, followingthe previously described objectives, the document is split in two parts.Part II presents a study of Flame Transfer Function (FTF) measurements using LES.Chapter 2 introduces this issue in its historical background and presents a short review of FTFlinear models in reacting flows.Chapter 3 focuses on FTF measurement techniques using LES. In particular, one describes a procedureinvolving the response of the flame to a filtered-white noise and using the inversion of the Wiener-Hopfrelation to get the FTF.Chapter 4 presents the results obtained using different methods for FTF measurements in four configurations.• First, the main aspects of FTF are obtained in a 1D laminar flame.• Then, a comparison of three methods of FTF is given in a laminar conical flame.• In a third section, a local comparison of two largely used methods is done in a 3D-turbulent forcedconfiguration. This section also provides a comparison of global time delays given by these methodswith experimental results.• Finally, a FTF method using filtered-white noise local response is compared to the classical harmonicmethod in a 3D-turbulent forced configuration.Part III presents a theoretical and numerical study of disturbance energies in combustion.Chapter 5 presents a short sample of the undergoing research effort towards the prediction of thermoacousticinstabilities in reacting flows. This chapter also introduces the numerical post-processing tooldeveloped here to investigate budgets of disturbance energies in flows.Chapter 6 focuses on the derivation of three disturbance energies.• The first two conservation equations are linear and respectively refer to Pressure-Velocity (PV)and Entropy disturbance energies.• The third equation is nonlinear and aims at giving the levels of disturbance energy in nonlinearcases.Chapter 7 focuses on the analysis of budgets of these disturbance energies in two configurations.• The first configuration is a 1D flame and gives access to the main aspects of the unsteady evolutionof disturbance energies.30

Objectives & Organisation• The second configuration is a 2D laminar flame. This case is harmonically forced and also madeunstable by increasing the outlet reflection coefficient. Results obtained in both cases are compared.The influence of the choice of the reference field is also investigated. A procedure forbalance analysis in turbulent cases using LES is proposed.Chapter 8 presents a validation of Rayleigh and Chu linear stability criteria in an unstable 2D laminarflame. These criteria fail to predict the instability and two new criteria are proposed which succeed.The first one extends the Rayleigh linear stability criterion and the second one aims at detecting thermoacousticinstabilities when no linearization of the flow is possible.31

OBJECTIVES & ORGANISATION32

Part IDescription of the numerical tool

Chapter 1Description of the numerical tool AVBPThis chapter focuses on some aspects of the numerical tool used for all simulations during this PhD.AVBP solver is a modern software tool for Computational Fluid Dynamics (CFD) of high flexibility,efficiency and modularity. It is an unstructured solver capable of handling grids of any cells type. Inthis work, both structured and unstructured grids have been used. AVBP solves the unsteady laminarand turbulent compressible Navier-Stokes equations in two and three space dimensions using numericalmethod which are second or third order in space and time.In the following sections, attention will be paid on to two important aspects of the code, the equationsand the main underlying hypothesis used for Direct Numerical Simulations (DNS) and the models usedfor Large Eddy Simulations (LES). Some information concerning boundary conditions and the implementationof artificial viscosity in AVBP may also be found.Finally, some details on the cell-vertex discretization are also provided. This section gives the main differencesbetween a ”central difference” and the Lax-Wendroff scheme that will be used in the third Partof this work.1.1 Governing Equations and hypothesis for DNSIn this section the compressible Navier-Stokes equations are described (as found in fundamental CFDtext books such as [2, 53] ) . This chapter focuses on unfiltered equations used in Direct NumericalSimultation. The LES extensions will be described on page 43.1.1.1 The governing equationsThroughout this document, the index notation is adopted for the description of the governing equations.Summation rule is henceforth implied over repeated indices (Einstein’s rule of summation). Note howeverthat index k is reserved to refer to the k th species and will not follow the summation rule unlesspecifically mentioned or implied by the ∑ sign.

DESCRIPTION OF THE NUMERICAL TOOL AVBPThe set of conservation equations describing the evolution of a compressible flow with chemicalreactions of thermodynamically active scalars reads,∂ρ E∂t∂ρ u i∂t+ ∂ (ρ u i u j ) = −∂ [P δ ij − τ ij ], (1.1)∂x j ∂x j+ ∂ (ρ E u j ) = −∂ [u i (P δ ij − τ ij ) + q j ] + ˙ω T + Q r , (1.2)∂x j ∂x j∂ρ k∂t + ∂ (ρ k u j ) = −∂ [J j,k ] + ˙ω k . (1.3)∂x j ∂x jIn Eqs 1.1-1.3 respectively corresponding to the conservation laws for momentum, total energy andspecies, the following symbols denote respecticely ρ, u i , E, ρ k , density, the velocity vector, the totalenergy per unit mass and ρ k = ρY k for k = 1 to N ( N is the total number of species). The source termin the total energy equation, Eq. 1.2s, is decomposed for convenience into a chemical source term anda radiative source term such that: S = ˙ω T + Q r . Corresponding source terms in the species transportequations, Eq. 1.3, are noted, ˙ω k .It is usual to decompose the flux tensor into an inviscid and a viscous component. They are respectivelynoted for the three conservation equations:Inviscid terms:⎛⎝ρ u i u j + P δ ij(ρE + P δ ij ) u jρ k u j⎞⎠ (1.4)where the hydrostatic pressure P is given by the equation of state for a perfect gas (Eq. 1.12).Viscous terms:The components of the viscous flux tensor take the form:⎛⎝−τ ij−(u i τ ij ) + q jJ j,k⎞⎠ (1.5)J j,k is the diffusive flux of species k and is presented in subsection on page 38 (Eq. 1.23). The stresstensor τ ij is explicited in subsection on page 39 (Eq. 1.24). Finally, subsection on page 39 is devoted tothe heat flux vector q j (Eq. 1.27).1.1.2 Thermodynamical variablesThe standard reference state used is P 0 = 1 bar and T 0 = 0 K. The sensible mass enthalpies (h s,k ) andentropies (s k ) for each species are tabulated for 51 values of the temperature (T i with i = 1...51) rangingfrom 0K to 5000K with a step of 100K. Therefore these variables can be evaluated by:36

Governing Equations and hypothesis for DNSh s,k (T i ) =∫ TiT 0 =0KC p,k dT = hm s,k (T i) − h m s,k (T 0)W k, and (1.6)s k (T i ) = sm k (T i) − s m k (T 0)W k, with i = 1, 51 (1.7)The superscript m corresponds to molar values. The tabulated values for h s,k (T i ) and s k (T i ) canbe found in the JANAF tables [146]. With this assumption, the sensible energy for each species can bereconstructed using the following expression :e s,k (T i ) =∫ TiT 0 =0KC v,k dT = h s,k (T i ) − r k T i i = 1, 51 (1.8)Note that the mass heat capacities at constant pressure c p,k and volume c v,k are supposed constantbetween T i and T i+1 = T i + 100. They are defined as the slope of the sensible enthalpy (C p,k =∂h s,k∂T) and sensible energy (C v,k = ∂e s,k∂T). The sensible energy henceforth varies continuously with thetemperature and is obtained by using a linear interpolation:e s,k (T ) = e s,k (T i ) + (T − T i ) e s,k(T i+1 ) − e s,k (T i )T i+1 − T i(1.9)The sensible energy and enthalpy of the mixture may then be expressed as:ρe s =ρh s =N∑N∑ρ k e s,k = ρ Y k e s,k (1.10)k=1k=1k=1N∑N∑ρ k h s,k = ρ Y k h s,k (1.11)k=11.1.3 The equation of stateThe equation of state for an ideal gas mixture writes:P = ρ r T (1.12)where r is the gas constant of the mixture dependant on time and space: r = R Wwhere W is the meanmolecular weight of the mixture:1NW = ∑ Y k(1.13)W kk=137

DESCRIPTION OF THE NUMERICAL TOOL AVBPThe gas constant r and the heat capacities of the gas mixture depend on the local gas composition as:r = R W = N ∑k=1C p =C v =Y kW kR =N∑Y k r k (1.14)k=1N∑Y k C p,k (1.15)k=1N∑Y k C v,k (1.16)k=1where R = 8.3143 J/mol.K is the universal gas constant. The adiabatic exponent for the mixture is givenby γ = C p /C v . Thus, the gas constant, the heat capacities and the adiabatic exponent are no longerconstant. Indeed, they depend on the local gas composition as expressed by the local mass fractionsY k (x, t):r = r(x, t), C p = C p (x, t), C v = C v (x, t), and γ = γ(x, t) (1.17)The temperature is deduced from the the sensible energy, using Eqs. 1.9 and 1.10. Finally boundaryconditions make use of the speed of sound of the mixture c which is given by:c 2 = γ r T (1.18)1.1.4 Conservation of Mass: Species diffusion fluxIn multi-species flows the total mass conservation implies that:N∑k=1Y k V ki = 0 (1.19)where Vik are the components in directions (i=1,2,3) of the diffusion velocity of species k. They areoften expressed as a function of the species gradients using the Hirschfelder Curtis approximation:X k V ki= −D k∂X k∂x i, (1.20)where X k is the molar fraction of species k : X k = Y k W/W k . In terms of mass fraction, the approximation1.20 may be expressed as:Y k V ki= −D kW kW38∂X k∂x i, (1.21)

Governing Equations and hypothesis for DNSSumming Eq. 1.21 over all k’s shows that the approximation 1.21 does not necessarily comply withequation 1.19 that expresses mass conservation. In order to achieve this, a correction diffusion velocity⃗V c is added to the convection velocity to ensure global mass conservation (see [107]) as:V ci =N∑k=1D kW kW∂X k∂x i, (1.22)and computing the diffusive species flux for each species k as:()W k ∂X kJ i,k = −ρ D k − Y k Vic , (1.23)W ∂x iHere, D k are the diffusion coefficients for each species k in the mixture (see subsection on page 40);J i,k . Using Eq. 1.23 to determine the diffusive species flux implicitly verifies Eq. 1.19.1.1.5 Viscous stress tensorThe stress tensor τ ij is given by the following relations:τ ij = 2µ(S ij − 1 3 δ ijS ll ), (1.24)andS ij = 1 2 (∂u j∂x i+ ∂u i∂x j), (1.25)Eq. 1.24 may also be written:τ xx = 2µ 3τ yy = 2µ 3τ zz = 2µ 3(2∂u∂x − ∂v∂y − ∂w∂z ),∂z ),(2∂v∂y − ∂u∂x − ∂w(2∂z∂w − ∂u∂x − ∂v∂y ),τ xy = µ( ∂u∂y + ∂v∂x )τ xz = µ( ∂u∂z + ∂w∂x )τ yz = µ( ∂v∂z + ∂w∂y ) (1.26)where µ is the shear viscosity (see subsection on page 40).1.1.6 Heat flux vectorFor multi-species flows, an additional heat flux term appears in the diffusive heat flux. This term is dueto heat transport by species diffusion. The total heat flux vector then writes:q i = −λ ∂TN∑()W k ∂X k−ρ D k − Y k Vic h s,k = −λ ∂T N∑+ J i,k h s,k , (1.27)∂x} {{ } iW ∂x i∂x ik=1k=1} {{ }Heat conductionHeat flux through species diffusionwhere λ is the heat conduction coefficient of the mixture (see subsection on page 40).39

DESCRIPTION OF THE NUMERICAL TOOL AVBP1.1.7 Transport coefficientsIn CFD codes for multi-species flows the molecular viscosity µ is often assumed to be independent ofthe gas composition and close to that of air 1 so that the classical Sutherland law can be used. In a firststep, the same assumption for the multi-gas AVBP solver is made, yielding:µ = c 1T 3/2T + c 2T ref + c 2T 3/2ref(1.28)where c 1 and c 2 must be determined so as to fit the real viscosity of the mixture. For air at T ref = 273 K,c 1 = 1.71e-5 kg/m.s and c 2 = 110.4 K (see [153]). These values are given by the user. A second law isavailable, called Power law:with b typically ranging between 0.5 and 1.0. For example b = 0.76 for air.µ = c 1 ( TT ref) b (1.29)The heat conduction coefficient of the gas mixture can then be computed by introducing the molecularPrandtl number of the mixture as:λ = µC pP r(1.30)with P r supposed as constant in time and space and is given by the user. The computation of the speciesdiffusion coefficients D k is a specific issue. These coefficients should be expressed as a function of thebinary coefficients D ij obtained from kinetic theory (Hirschfelder et al. [54]). The mixture diffusioncoefficient for species k, D k , is computed as (Bird et al. [11]):D k =1 − Y k∑ Nj≠k X j/D jk(1.31)The D ij are complex functions of collision integrals and thermodynamic variables. For a DNS codeusing complex chemistry, using Eq. 1.31 makes sense. However in most cases, DNS uses a simplifiedchemical scheme and modeling diffusivity in a precise way is not needed so that this approach is muchless attractive. Therefore a simplified approximation is used in AVBP for D k . The Schmidt numbersS c,k of the species are supposed to be constant so that the binary diffusion coefficient for each species iscomputed as:D k = µρ S c,k(1.32)Note that the Schmidt number for each species k is assumed to be constant in time and space and isgiven by the user. P r and S c,k model the laminar (thermal and molecular) diffusion. Usual values ofSchmidt and Prandtl numbers for premixed flames are those given by PREMIX in the burnt gas.1 This introduces errors that are less important than those related to the thermodynamic properties.40

Governing Equations and hypothesis for DNS1.1.8 KineticsThe source term on the right hand side of Eqs. 1.2 & 1.3 respectively writes:( ˙ωT˙ω k)where ˙ω T is the rate of heat release and ˙ω k the reaction rate of species k. Source terms in the momentumequation, Eq. 1.1, may also appear and are in this case used to impose momentum components, forexample an imposed pressure gradients in periodic flows. In most cases however, they are null.The combustion model of AVBP is an Arrhenius law written for N reactants M k and for M reactionsas:N∑N∑ν kj ′ M kj ⇋ ν kj ′′ M kj, j = 1, M (1.33)k=1The reaction rate of species k ( ˙ω k ) is the sum of rates ˙ω kj produced by all M reactions:˙ω k =k=1M∑ ∑M˙ω kj = W k ν kj Q j (1.34)where ν kj = ν ′′kj − ν′ kj and Q j is the rate progress of reaction j and is written:Q j = K f,jj=1N ∏k=1j=1( ρY k) ν′ kj − K r,jW kK f,j and K r,j are the forward and reverse rates of reaction j:N ∏k=1( ρY k) ν′′ kj (1.35)W kK f,j = A f,j exp(− E a,jRT ) (1.36)where A f,j and E a,j are the pre-exponential factor and the activation energy given in the input file. K r,jis deduced from the equilibrium assumption:where K eq is the equilibrium constant defined by Kuo [70]:K eq =K r,j = K f,jK eq(1.37)( p0) P (Nk=1 ν kj ∆S0jexpRTR− ∆H0 jRT)(1.38)where p 0 = 1 bar. ∆Hj 0 and ∆S0 jchanges for reaction j:are respectively the enthalpy (sensible + chemical) and the entropyN∑∆Hj 0 = h j (T ) − h j (0) = ν kj W k (h s,k (T ) + ∆h 0 f,k ) (1.39)k=141

DESCRIPTION OF THE NUMERICAL TOOL AVBP∆S 0 j =N∑ν kj W k s k (T ) (1.40)k=1where ∆h 0 f,k is the mass enthalpy of formation of species k at temperature T 0 = 0 K and is given in theinput file (molar value).The heat release is calculated as:N∑˙ω T = − ˙ω k ∆h 0 f,k (1.41)k=142

1.2 Governing equations for LES1.2 Governing equations for LES1.2.1 The LES ConceptLarge Eddy Simulation (LES) [122, 116] is nowadays recognized as an intermediate approach in comparisonsto the more classical Reynolds Averaged Navier-Stokes (RANS) methodologies. Althoughconceptually very different these two approaches aim at providing new systems of governing equationsto mimic the characteristics of turbulent flows.The derivation of the new governing equations is obtained by introducing operators to be applied tothe set of compressible Navier-Stokes equations. Unclosed terms arise from these manipulations andmodels need to be supplied for the problem to be solved. The major differences between RANS andLES come from the operator employed in the derivation. In RANS the operation consists of a temporalor ensemble average over a set of realizations of the studied flow [116, 22]. The unclosed terms arerepresentative of the physics taking place over the entire range of frequencies present in the ensembleof realizations under consideration. In LES, the operator is a spatially localized time independent filterof given size, △, to be applied to a single realization of the studied flow. Resulting from this ”spatialaverage” is a separation between the large (greater than the filter size) and small (smaller than the filtersize) scales. The unclosed terms are in LES representative of the physics associated with the smallstructures (with high frequencies) present in the flow. Figure 1.1 illustrates the conceptual differencesbetween (a) RANS and (b) LES when applied to a homogeneous isotropic turbulent field.(a)(b)Figure 1.1 - Conceptual representation of (a) RANS and (b) LES applied to a homogeneous isotropic turbulentfield.Due to the filtering approach, LES allows a dynamic representation of the large scale motions whosecontributions are critical in complex geometries. The LES predictions of complex turbulent flows arehenceforth closer to the physics since large scale phenomena such as large vortex shedding and acousticwaves are embedded in the set of governing equations [107].43

DESCRIPTION OF THE NUMERICAL TOOL AVBPFor the reasons presented above, LES has a clear potential in predicting turbulent flows encounteredin industrial applications. Such possibilities are however restricted by the hypothesis introduced whileconstructing LES models.This chapter describes the equation solved for LES of reacting flows in AVBP. First, the filtered equationssolved by AVBP for a turbulent non-reacting flow are described (subsection on page 44). Subsectionon page 47 presents the models used for turbulent viscosity. Subsection on page 50 describes specificallythe models for flame/turbulence interactions (the TF model) and shows how this model is coupled to thefiltered equations.1.2.2 The Governing Equations for Non-Reacting FlowsThe filtered quantity f is resolved in the numerical simulation whereas f ′ = f −f is the subgrid scale partdue to the unresolved flow motion. For variable density ρ, a mass-weighted Favre filtering is introducedsuch as:ρ ˜f = ρf (1.42)The balance equations for large eddy simulations are obtained by filtering the instantaneous balanceequations 1.1, 1.2 and 1.3:∂ρ ũ i∂t+ ∂ (ρ ũ i ũ j ) = −∂ [P δ ij − τ ij − τ t ij ], (1.43)∂x j ∂x j∂ρ Ẽ∂t+ ∂ (ρ∂x Ẽ ũ j) = −∂ [u i (P δ ij − τ ij ) + q j + q t j ] + ˙ω T + Q r , (1.44)j ∂x j∂ρ Ỹk∂t+ ∂ (ρ∂x Ỹk ũ j ) = −∂t[J j,k + J j,k ] + ˙ωk , (1.45)j ∂x jwhere a repeated index implies summation over this index (Einstein’s rule of summation). Note alsothat throughout the document, the index k is reserved to refer to the k th species and does not follow thesummation rule (unless specifically mentioned).For generality, the filtered chemical source terms are mentioned here if a user wants to implement anew combustion model. For the standard AVBP models for flame / turbulence interactions (TF and DTF),a specific implementation is done (see Section on page 50). The cut-off scale corresponds to the meshsize (implicit filtering). As usually done, we assume that the filter operator and the partial derivativecommute.In Eqs. 1.43, 1.44 and 1.45, the flux tensor can be divided in three parts: the inviscid part, the viscouspart and the subgrid scale turbulent part.Inviscid terms:The three spatial components of the inviscid flux tensor are the same as in DNS but based on thefiltered quantities:44

Governing equations for LES⎛⎝ρũ i ũ j + P δ ijρẼũ j + P u j δ ijρ k ũ j⎞⎠ (1.46)Viscous terms:The components of the viscous flux tensor take the form:⎛⎝−τ ij−(u i τ ij ) + q jJ j,k⎞⎠ (1.47)Filtering the balance equations leads to unclosed quantities, which need to be modeled.Subgrid scale turbulent terms:The components of the turbulent subgrid scale flux take the form:⎛−τt ij⎞⎝ qt jtJ j,k⎠ (1.48)The filtered viscous terms in non reactive flowsThe filtered diffusion terms are (see T. Poinsot and D. Veynante, Chapter 4 [107]) :• the laminar filtered stress tensor ˜τ ij is given by the following relations:andEq. 1.49 may also be written:τ xx ≈ 2µ 3τ yy ≈ 2µ 3τ zz ≈ 2µ 3τ ij = 2µ(S ij − 1 3 δ ijS ll ),≈ 2µ( ˜S ij − 1 3 δ ij ˜S ll ),(1.49)˜S ij = 1 2 (∂ũ j∂x i+ ∂ũ i∂x j), (1.50)(2∂eu∂x − ∂ev∂y − ∂ ew∂z ),∂z ),(2∂ev∂y − ∂eu∂x − ∂ ew(2∂ ew∂z − ∂eu∂x − ∂ev∂y ),τ xy ≈ µ( ∂eu∂y + ∂ev∂x )τ xz ≈ µ( ∂eu∂z + ∂ ew∂x )τ yz ≈ µ( ∂ev∂z + ∂ ew∂y ) (1.51)• the filtered diffusive species flux vector is:(= −ρJ i,k≈ −ρWD k ∂X k k W∂X e k∂x i(D kW kW∂x i− Y k V ic )− ỸkṼic ) ,(1.52)45

DESCRIPTION OF THE NUMERICAL TOOL AVBPwhere higher order correlations between the different variables of the expression are assumednegligible.• the filtered heat flux is :q i= −λ ∂T∂x i+ ∑ Nk=1 J i,kh s,k≈ −λ ∂ T e∂x i+ ∑ Nk=1 J i,k ˜h(1.53)s,kThese forms assume that the spatial variations of molecular diffusion fluxes are negligible and can bemodelled through simple gradient assumptions.Subgrid scale turbulent terms for non-reacting LESAs highlighted above, filtering the transport equations yields a closure problem evidenced by the so called”Sub-Grid Scale” (SGS) turbulent fluxes (see Eq. on page 44). For the system to be solved numerically,closures need to be supplied. Details on the forms and models available in AVBP are given in thissubsection.• the Reynolds tensor is :τ ij t = −ρ(ũ i u j − ũ i ũ j ), (1.54)τ ij t is modeled as:τ ij t = 2 ρ ν t ( ˜S ij − 1 3 δ ij ˜S ll ), (1.55)The modelisation of ν t is explained in subsection on page 47.• the subgrid scale diffusive species flux vector:J i,kt= ρ ( ũ i Y k − ũ i Ỹ k ), (1.56)J i,ktis modeled as:J i,kt= −ρ(D kt W kW∂ ˜X k∂x i)− ỸkṼic,t, (1.57)withD t k =ν tS t c,k(1.58)The turbulent Schimdt number Sc,k t = 1 is the same for all species and is fixed in the source code(like Pr). t Note also that having one turbulent Schmidt number for all the species does not imply,Ṽ c,t = 0 because of the W k /W term in Eq. 1.57.46

Governing equations for LES• the subgrid scale heat flux vector:q i t = ρ(ũiE − ũ i Ẽ), (1.59)where e is the sensible energy. In the source code, the modelisation for ˜q t is written :withq i t = −λ t∂ ˜T∂x i+N∑J i,kt ˜hs,k , (1.60)k=1The turbulent Prandtl number is fixed in AVBP. Usually, P t r = 0.9.λ t = µ tC pPrt . (1.61)The correction diffusion velocities are henceforth obtained in AVBP from:()N∑Ṽic + Ṽ c,t µi=+µ t W k ∂ ˜X k, (1.62)ρS c,k W ∂x iand where Eqs. 1.32 and 1.58 are used.k=1ρS t c,k1.2.3 Models for the subgrid stress tensor τ ij t :In AVBP, three models are available:• the Smagorinsky model,• the Wall Adapting Linear Eddy (WALE) model,• the Filtered Smagorinsky model.A description of the characteristics of each model is given in the next subsection.These LES models are derived on the theoretical ground that the LES filter is spatially and temporallyinvariant. Variations in the filter size due to non-uniform meshes or moving meshes are not directlyaccounted for in the LES models. Change of cell topology is only accounted for through the use of thelocal cell volume, that is △ = V 1/3cell .Use of artificial viscosity is in theory prohibited in LES. It should therefore be used with a lot ofcaution and in no way overshadow the LES model contribution.The filtered compressible Navier-Stokes equations exhibit sub-grid scale (SGS) tensors and vectorsdescribing the interaction between the non-resolved and resolved motions. The influence of the SGS47

DESCRIPTION OF THE NUMERICAL TOOL AVBPon the resolved motion is taken into account in AVBP by a SGS model based on the introduction of aturbulent viscosity, ν t . Such an approach assumes the effect of the SGS field on the resolved field to bepurely dissipative.The previous hypothesis is essentially valid within the cascade theory of turbulence. The notion ofturbulent viscosity can therefore be introduced and yields a general model for the SGS which readsτ ijt= −ρ (ũ i u j − ũ i ũ j )with= 2 ρ ν t ˜Sij − 1 3 τ ll t δ ij , (1.63)˜S ij = 1 2( ∂ũi∂x j+ ∂ũ j∂x i)− 1 3∂ũ k∂x kδ ij . (1.64)In Eqn. (1.63) τ ij t is the SGS tensor to be modeled, ν t is the SGS turbulent viscosity, ũ i is the Favrefiltered velocity vector (compressible flows) and ˜S ij is the resolved strain rate tensor. The three modelsin AVBP only differ through the estimation of ν t whose expressions are given below.Smagorinsky modelν t = (C S △) 2 √2 ˜S ij ˜Sij , (1.65)where △ denotes the filter characteristic length (cube-root of the cell volume), C S is the model constantset to 0.18 but can vary between 0.1 and 0.18 depending on the flow configuration. The Smagorinskymodel [141] was developed in the sixties and heavily tested for multiple flow configurations. This closurehas the particularity of supplying the right amount of dissipation of kinetic energy in homogeneousisotropic turbulent flows. Locality is however lost and only global quantities are maintained. It is knownas being ”too dissipative” and transitioning flows are not suited for its use [122].Note: in AVBP, the expression for ν t is not exactly as given in (1.65). Rather a simplified version of˜S ij (as shown in (1.64) ) is used; compressibility effects are neglected and ∂ũ k /∂x k ≈ 0.WALE models d ij = 1 2 (˜g2 ij + ˜g 2 ji) − 1 3 ˜g2 kk δ ij, (1.66)ν t = (C w △) 2 (s d ij sd ij )3/2( ˜S, (1.67)ij ˜Sij ) 5/2 +(s d ij sd ij)5/4where △ denotes the filter characteristic length (cube-root of the cell volume), C w = 0.4929 is the modelconstant and ˜g ij denotes the resolved velocity gradient. The WALE model [40] was developed for wallbounded flows in an attempt to recover the scaling laws of the wall. Similarly to the Smagorinsky modellocality is lost and only global quantities are to be trusted.48

Governing equations for LESFiltered Smagorinsky modelν t = (C SF △) 2 √2 HP ( ˜S ij ) HP ( ˜S ij ), (1.68)where △ denotes the filter chracteristic length (cube-root of the cell volume), C SF = 0.37 is the modelconstant and HP ( ˜S ij ) denotes the resolved strain rate tensor obtained from a high-pass filtered velocityfield. This model was developed in order to allow a better representation of local phenomena typical ofcomplex turbulent flows [40]. With the Filtered Smagorinsky model transition is better predicted andlocality is in general better preserved.49

DESCRIPTION OF THE NUMERICAL TOOL AVBP1.3 The Thickened Flame (TF) model for LESA difficult problem is encountered for Large Eddy Simulation of premixed flames: the thickness δ 0 L of apremixed flame is generally smaller than the standard mesh size ∆ x used for LES. For this reason, theThickened Flame (TF) model has been developed so as to resolve the flame fronts on a LES mesh. Inturbulent flows, the interaction between turbulence and chemistry is altered: eddies smaller than F δ 0 L donot interact with the flame any longer. As a result, the thickening of the flame reduces the ability of thevortices to wrinkle the flame front. As the flame surface is reduced, the reaction rate is underestimated.In order to correct this effect, an efficiency function E has been developed [28] from DNS results andimplemented into AVBP (see Fig. 1.2). It is described in the next subsection.Figure 1.2 - Direct Numerical Simulation of flame/turbulence interactions by Veynante ([4], [104]). Left: nonthickened flame, right: thickened flame (F = 5).1.3.1 The combustion subgrid scale model: EA complete description of the efficiency function is given in ref [28]. The underlying model philosophycan be summarized through 3 main steps:• The wrinkling factor of the flame surface Ξ is estimated from the flame surface density Σ, assumingan equilibrium between the turbulence and the subscale flame surface:Ξ ≃ 1 + α ∆ eSL0 〈a T 〉 s (1.69)where 〈a T 〉 s is the subgrid scale strain rate, ∆ e is the filter size and α is a model constant.• 〈a T 〉 s is estimated from the filter size ∆ e and the subgrid scale turbulent velocity u ′ ∆ e: 〈a T 〉 s =Γu ′ ∆ e/∆ e . The function Γ corresponds to the integration of the effective strain rate induced by50

The Thickened Flame (TF) model for LESall scales affected by the artificial thickening, i.e. between the Kolmogorov η K and the filter ∆ escales (see also [84]). Γ is written as:( ) [] (∆e )∆eΓδL1 , u′ 2∆ e1.23SL0 = 0.75 exp −( )u′∆e/SL0 0.3δL1 (1.70)Finally, the efficiency function is defined as the wrinkling ratio between the non-thickened referenceflame and the thickened flame:(∆eE = Ξ(δ0 L ) 1 + αΓΞ(δL 1 ) = δL( 01 + αΓ ∆eδ 1 L), u′ ∆e u ′∆eSL0 SL0, u′ ∆eS 0 L) u ′∆eS 0 L(1.71)SL 0 and δ0 Lare the laminar flame speed and the laminar flame thickness, respectively, when F = 1and δL 1 = F δ0 L .E varies between 1 (weak turbulence) to E max ≃ F 2/3 (large wrinkling at the subgrid scale). Inturbulent premixed zones, the efficiency function is determined to ensure that the turbulent flamespeed will be E SL 0 = S T . The efficiency function is required when the vortex size r is definedby δL 0 > r > δc L for a real flame and by δ1 L = β F δ0 L > r > δc L for a thickened flame. δc L isa cut-off length scale: for vortices lower than δL c , the flame remains unaffected. δc Lis defined in[28], Eq. 31.• The filter size ∆ e corresponds to the greatest scale affected by the flame thickening, that is tosay δL 1 . In practice, ∆ e = 10∆ x with ∆ x = 3√ voln. The subgrid scale turbulent velocity u ′ ∆ eis estimated using the operateur OP 2 , based on the rotational of the velocity field to remove thedilatational part of the velocity which must not be counted as ”turbulence”. A Laplacian operatoris directly applied to the velocity :u ′ ∆ e= c 2 ∆ 3 ∂ 2 ( )∂u nx| ɛ lmn | (1.72)∂x j ∂x j ∂x mwith c 2 ≈ 2 and where ɛ lmn stands for the permutation tensor.Estimation of the model constant αThe model constant α is estimated to match the asymptotic behavior of the wrinkling factor Ξ versusRMS velocity u’ for thin flames when ∆ e goes to the integral length scale l t , the flame wrinkling Ξ goesto Ξ max defined by:Ξ max = 1 + βu ′ /S 0 L (1.73)with u’ the velocity at length scale l t . α is then deduced from Eq. 1.73:α = β2 ln(2)3c ms [Re 1/2t − 1](1.74)where Re t = u′ l tνis the turbulent Reynolds number and c ms = 0.28. The reader is referred to [28] formore details.51

DESCRIPTION OF THE NUMERICAL TOOL AVBPThe TFLES1 model: In this method, the user has to provide an evaluation for α. For that, Re t = u′ l tνhas to be evaluated. Typically, l t is of the order of the largest vortex in the computation domain. However,strong variations of Re t may be observed whether the vortex is near the injectors or in the burner, in thefresh gas or in the burnt gas. In usual combustion chamber, Re t varies between 100 and 100,000, then αvaries between 0.165 and 0.0052 (assuming β is of the order of unity).The TFLES2 model: The objective of this model is to compute Re t as a function of space andtime and to give a new formulation for α. The different steps to obtain the efficiency function are thefollowing:• First, the integral length scale is estimated. Near the wall, the log-law in turbulent channel flowis used to estimate l t : l t ≈ 0.4d w , where d w is the distance to the wall. Numerically, d w is theminimum distance from walls to the considered node. Far from walls, the user must provide anestimation of the integral length scale l 0 t in the whole chamber. Usually l 0 t is of the order of 1/10to 1/4 of the transversal size of the burner. Finally, the local integral length scale is:l t ≈ min(0.4d w , l 0 t ) (1.75)• Assuming a Kolmogorov cascade with constant transfer rate ɛ, the fluctuation velocity u’ is deducedby:ɛ = u′3l t= u′3 ∆ e∆ e(1.76)• The subgrid scale velocity u ′ ∆ eis overestimated when using Eq. 1.72 close to the wall, that is tosay in boundary layers. For this reason a damping function depending on the distance to the wallis used and u ′ ∆ eis then determined as:u ′ ∆ e=(1 − exp(− d w0.2δ 1 L))c 2 ∆ 3 ∂ 2 ( )∂u nx| ɛ lmn | (1.77)∂x j ∂x j ∂x mNear the wall, l t goes to zero while the filter scale ∆ e is nearly constant. When ∆ e < l t , theKolmogorov assumption is not valid any more, that is why ∆ e and u ′ ∆ eare replaced by l t and u’.Such an approximation is acceptable, because most of the combustion occurs in the center of thechamber.• A correction for the turbulent Reynolds number is done wherever the cut-off length scale δL c issmaller or greater than the Kolmogorov scale η k . If δL c < η k, η k is the smallest efficient structuresize and δL c is irrelevant. The Reynolds number is then Re t = u′ l tν. Recalling that the flameReynolds number is defined by: Re f = S0 L δ0 Lν≈ 1, and that the Kolmogorov scale Reynoldsnumber is written as: Re(η k ) = u′ (η k )η k= 1, the turbulent Reynolds number may be(ltν≈ u′ (η k )η kS 0 L δ0 L) 4/3.written: Re t =η kIf δcL> η k , the smallest efficient structure size is δL c and the Reynolds( )number estimation should be corrected with a function θ, such as: θ 2 Re t = lt 4/3.δ The finalL cmodel for α is then written:2ln(2)α = β3c ms [θ(Re t ) 1/2 (1.78)− 1]52

The Thickened Flame (TF) model for LES(ηkδ c Lθ = min((ηkδ c L) 2/3, 1) (1.79)⎛ ⎞) 2/3 ( ) 2/3= (Re t ) 1/2 lt ln(2) ⎜exp ⎝ −1.2 ⎟( )2c ms u ′ 0.3 ⎠ (1.80)The model constant β has to be fixed by the user in the input file and is of the order of 0.3.δ 0 LOther forms of efficiency function have been derived by Charlette and Meneveau [20, 21] but are notimplemented in AVBP yet.S 0 L1.3.2 Implementation of the standard Thickened Flame (TF) modelThe filtered equations for total energy and for species (Eqs. 1.44 & 1.45) must be modified in reactiveflows when the TF or DTF model is used. In this case, only the filtered equations for velocities (Eq. 1.43)are unchanged. For the species and energy, the filtered equations are replaced by the thickened equationsas follows:Viscous terms• the filtered diffusive species flux vector is given by:J i,k = − E F µ W k ∂ ˜X k+ ρS c,k W ∂x k Ṽi c , (1.81)iwith• the filtered heat flux is:Ṽ ci= E FN∑k=1q i = − E F µC pP rµ W k ∂ ˜X k, (1.82)ρS c,k W ∂x i∂ ˜T∂x i+N∑J i,k ˜hs,k , (1.83)k=1The source termThe filtered source term vector of Eq. 1.42 is written:(E ˙ωT ( e Y k , e T )FE ˙ω k ( e Y k , e T )F), (1.84)where ˙ω T (Ỹk, ˜T ) and ˙ω k (Ỹk, ˜T ) are reaction rates computed with the Arrhenius expression and thefiltered values of Y k and T . Note that this model should be used only for perfectly premixed cases since53

DESCRIPTION OF THE NUMERICAL TOOL AVBPmixing in the fresh gases, for example, is modified by thickening and not correctly handled with thefiltered terms of Eqs. 1.44 & 1.45. The actual transport equations for the TF model are summarizedbelow.Use of the TF model implies the following relation for the correction diffusion velocities:Ṽ ci+ Ṽ c,ti=N∑k=1E F µ W k ∂ ˜X k, (1.85)ρS c,k W ∂x i1.3.3 Final equations solved for the TF modelThe final set of LES equations solved for when performing LES of reacting flows with theTF modelfinally reads:∂ρ ũ i∂t∂ρ Ẽ∂t∂ρ Ỹk∂t+ ∂ (ρ ũ i ũ j ) = − ∂ [P δ ij − 2 (µ + µ t ) (∂x j ∂x ˜S ij − 1/3 ˜S]ll δ ij ) ,j+ ∂ (ρ∂x Ẽ ũ j) = − ∂ [ũ i P δ ij − 2 µ ũ i (j ∂x ˜S ij − 1/3 ˜S]ll δ ij )j[+ ∂ C p E F µ ∂ ˜T]∂x j P r ∂x j+ ∂∂x j(ρ Ỹk ũ j ) =+ ∂∂x j[ N∑+ E ˙ω TF ,k=1(E F µ W k ∂ ˜X)k− ρS c,k W ∂x Ỹk (Ṽ j c + Ṽ c,tj)j[∂E F µ W k ∂ ˜X k− ρ∂x j S c,k W ∂x Ỹk (Ṽ j c + Ṽ c,tj)j+ E ˙ω kF , 54]˜hs,k]

1.4 General aspects of the Boundary Conditions in AVBP1.4 General aspects of the Boundary Conditions in AVBPBoundary Conditions are an essential part in any CFD code, and especially in AVBP because of acousticspresent in the governing equations [127],[104]. The time integration in AVBP is performed with a multistageRunge-Kutta (RK) method. For simplicity, we only consider here a single-stage RK, which isactually a simple Euler integration, eventually combined with a Lax-Wendroff method for stability.Knowing the solution w n at time t, the solution w n+1 at time t + ∆t is computed for each node i as:w n+1i= w n i − ∆tV i.dw n i (1.86)where wi n = w(t, ⃗x i ) and win+1 = w(t + ∆t, ⃗x i ); ⃗x i is the coordinate vector, V i is the nodal volumearound node i and dwi n is the nodal residual at node i, as computed by the numerical scheme.The n superscript is here to remind that AVBP is an explicit code, hence this nodal residual only dependson quantities known at the timestep n.This formula is applied to each node of the computational domain (Ω). If no physical BC was imposed,the computed solution at each timestep would only depend on the initial solution and on the numericalscheme.In order to impose a BC on the border of the domain (∂Ω), we write:⎧⎨⎩w n+1i= w n i − ∆tV i· (dw n i ) scheme ∀x i ∈ Ω/∂Ωw n+1i= w n i − ∆tV i· (dw n i ) BC ∀x i ∈ ∂Ω(1.87)For each node lying on the boundary, we have replaced the scheme-predicted residual by a BC-correctedresidual. This operation is known as imposing the BC in a “hard way”. A “weak” method is additionallyused in conjunction with this “hard” method for certain BC. In this case, some gradients and fluxes aremodified before applying the numerical scheme. It is mainly used for “Von Neuman” like conditions, asadiabaticity or impermeability.The AVBP BC can be classified through two categories:• Non-characteristic BC, that work directly on conserved variables,• Characteristic BC, that use a wave decomposition to modify residuals.55

DESCRIPTION OF THE NUMERICAL TOOL AVBP1.5 Artificial Viscosity in AVBPThe numerical discretization methods in AVBP are spatially centered. These types of schemes are knownto be naturally subject to small-scale oscillations in the vicinity of steep solution variations. This iswhy it is common practice to add a so-called artificial viscosity (AV) term to the discrete equations, toavoid these spurious modes (also known as “wiggles”) and in order to smooth very strong gradients.We describe here the different AV methods used in AVBP. These AV models are characterized by the“linear preserving” property which leaves unmodified a linear solution on any type of element. Themodels are based on a combination of a “shock capturing” term (called 2 nd order AV) and a “backgrounddissipation” term (called 4 th order AV). In AVBP, adding AV is done in two steps:• first a sensor detects if AV is necessary, as a function of the flow characteristics,• then a certain amount of 2 nd and 4 th AV is applied, depending on the sensor value and on userdefinedparameters.1.5.1 The sensorsA sensor ζ Ωj is a scaled parameter which is defined for every cell Ω j of the domain that takes values fromzero to one. ζ Ωj = 0 means that the solution is well resolved and that no AV should be applied whileζ Ωj = 1 signifies that the solution has strong local variations and that AV must be applied. This sensoris obtained by comparing different evaluations (on different stencils) of the gradient of a given scalar(pressure, total energy, mass fractions, . . . ). If these gradients are identical, then the solution is locallylinear and the sensor is zero. On the contrary, if these two estimations are different, local non-linearitiesare present, and the sensor is activated. The key point is to find a suitable sensor-function that is non-zeroonly at places where stability problems occur.The ‘Colin-sensor’ (ζΩ C j) [28] used in this thesis is described here.The Colin sensorFor most unsteady turbulent computations it is necessary to have a sharp sensor, which is very smallwhen the flow is sufficiently resolved, and which is nearly maximum when a certain level of nonlinearitiesoccurs.S is the scalar quantity the sensor is based on (usually S is the pressure) ∆ k 1 and ∆k 2 functions aredefined as:∆ k 1 = S Ωj − S k ∆ k 2 = ( ∇S) ⃗ k .(⃗x Ωj − ⃗x k ) (1.88)where a k subscript denotes cell-vertex values while Ω j is the subscript for cell-averaged values.( ∇S) ⃗ k is the gradient of S at node k as computed in AVBP.∆ k 1 measures the variation of S inside the cell Ω j (using only quantities defined on this cell). ∆ k 2 is an56

Artificial Viscosity in AVBPestimation of the same variation but on a wider stencil (using all the neighbouring cell of the node k).the aim of the so-called Colin-sensor is to preserve sharp gradients in the computation. its propertiescan be summarized as follows:• ζΩ C jis very small when both ∆ k 1 and ∆k 2 are small compared to S Ω j. This corresponds to lowamplitude numerical errors (when ∆ k 1 and ∆k 2 have opposite signs) or smooth gradients that arewell resolved by the scheme (when ∆ k 1 and ∆k 2 have the same sign).• ζΩ C jis small when ∆ k 1 and ∆k 2 have the same sign and the same order of magnitude, even if theyare quite large. This corresponds to stiff gradients well resolved by the scheme.• ζΩ C jis big when ∆ k 1 and ∆k 2 have opposite signs and one of the two term is large compared to theother. This corresponds to a high-amplitude numerical oscillation.• ζ C Ω jis big when either ∆ k 1 or ∆k 2 is of the same order of magnitude as S Ω j. This corresponds to anon-physical situation that originates from a numerical problem.The exact definition of the Colin-sensor is:ζΩ C j= 1 ( ( )) Ψ − Ψ01 + tanh− 1 ( ( )) −Ψ01 + tanh2δ 2δ(1.89)with:(∆ k )Ψ = max 0,k∈Ω j |∆ k ζkJ | + ɛ 1 S k∆ k = |∆ k 1 − ∆ k 2| − ɛ k max( )|∆ k 1|, |∆ k 2|(ɛ k max ( |∆ k 1= ɛ 2 1 − ɛ |, |∆k 2 |))3|∆ k 1 | + |∆k 2 | + S k(1.90)(1.91)(1.92)The numerical values used in AVBP are:Ψ 0 = 2.10 −2 δ = 1.10 −2 ɛ 1 = 1.10 −2 ɛ 2 = 0.95 ɛ 3 = 0.5 (1.93)WARNING:Note, that these definitions of Ψ and ɛ k apply only for the Navier-Stokes variables. For species, thereference value is not S k but 1, which is the maximum value of a species mass fraction:(∆ k )Ψ = max 0,k∈Ω j |∆ k ζkJ | + ɛ 1and ɛ k = ɛ 2(max ( |∆ k 11 − ɛ |, |∆k 2 |))3|∆ k 1 | + |∆k 2 | + 1(1.94)57

DESCRIPTION OF THE NUMERICAL TOOL AVBP1.5.2 The operatorsThere are two AV operators in AVBP: a 2 nd order operator and a 4 th order operator. All AV models inAVBP are a blend of these two operators. These operators have the following properties:• 2 nd order operator: it acts just like a “classical” viscosity. It smoothes gradients, and introducesartificial dissipation. It is thus associated to a sensor which determines where it must be applied.Doing this, the numerical scheme keeps its order of convergence in the zones where the sensor isinactive, while ensuring stability and robustness in the critical regions. Historically, it was used tocontrol shocks, but it can actually smooth any physical gradient.• 4 th order operator: it is a less common operator. It acts as a bi-Laplacian and is mainly used tocontrol spurious high-frequency wiggles.The way they are combined is determined both by the sensor and by user-defined parameters (smu2 andsmu4).Both operator contributions are first computed on each cell vertex, and are then scattered back to nodes(there is no divergence here, as it is done directly during the scattering operation).The 2 nd order operatorA cell contribution of the 2 nd order AV is first computed on each vertex of the cell Ω j :R k∈Ωj = − 1 N vV Ωj∆t Ωjsmu2 ζ Ωj (w Ωj − w k ) (1.95)The nodal residual is then found by adding the surrounding cells contributions:dw k = ∑ jR k∈Ωj (1.96)For example, on a 1D uniform mesh, of mesh size ∆x, and for ζ Ωj = ζ = cste :which can be interpreted as:with:ν AV =smu2 ζ ∆x22∆t=dw k = − smu2 ζ2dw k = −ν AV ∫smu2 ζ ∆x|u + c|2 CFL∆x∆t (w k−1 − 2w k + w k+1 ) (1.97)(∆ F k,∆x D w) dx (1.98)and∆ F Dk,∆x w = w k−1 − 2w k + w k+1∆x 2 (1.99)where ∆ F k,∆x D is exactly the classical FD Laplacian operator evaluated at k and of size ∆x.This shows that ν AV can be seen as an “artificial” viscosity (it has the same units as a physical viscosity),which is controlled by the user-defined parameter smu2. The smu2 parameter is therefore dimensionless.58

Artificial Viscosity in AVBPThe 4 th order operatorThe technique used for the 4 th order operator is identical to the technique of the 2 nd order operator. Acell contribution is first computed on each vertex:R k∈Ωj = 1 V [Ωjsmu4 (N v ∆t ⃗ ]∇w) Ωj · (⃗x Ωj − ⃗x k ) − (w Ωj − w k )ΩjThe nodal value is then found by adding every surrounding cells contributions:(1.100)dw k = ∑ jR k∈Ωj (1.101)For example, on a 1D uniform mesh, of mesh size ∆x, this yields:R k∈Ωleft = smu4 [(∆x 12 ∆t 2 (w k − w k−2+ w ) (k+1 − w k−1 −∆x) ·2∆x 2∆x2R k∈Ωright = smu4 [(∆x 12 ∆t 2 (w k+1 − w k−1+ w ) (k+2 − w k ∆x) ·2∆x 2∆x 2Adding these 2 contributions gives:which can be interpreted:with:) (wk−1 + w k−2)−− w k)]( (1.102) )]wk + w k+1− w k2(1.103)dw k = smu4 ∆x16∆t (w k−2 − 4w k−1 + 6w k − 4w k+1 + w k+2 ) (1.104)dw k = κ AV ∫(∆∆ F k,∆x D w) dx (1.105)κ AV = smu4.∆x416∆t= smu4.∆x3 |u + c|16 CFLand∆∆ F Dk,∆x w = w k−2 − 4w k−1 + 6w k − 4w k+1 + w k+2∆x 4 (1.106)where ∆∆ F k,∆x D is exactly the classical FD bi-Laplacian operator evaluated at k and of size ∆x.This shows that κ AV can be seen as an “artificial” 4 th order hyper-viscosity, which is controlled by theuser-defined parameter smu4. Just like smu2, the smu4 parameter is dimensionless.59

DESCRIPTION OF THE NUMERICAL TOOL1.6 Cell-Vertex DiscretizationThe flow solver used for the discretization of the governing equations is based on the “finite volume”(FV) method. There are two common techniques for implementing FV methods: the so called cellvertexand the cell-centered formulation. In the latter, not used in AVBP, discrete solution values arestored at the center of the control volumes (or grid cells), and neighbouring values are averaged acrosscell boundaries in order to calculate fluxes. The alternative cell-vertex technique, used as underlyingnumerical discretization method of AVBP, the discrete values of the conserved variables are stored at thecell vertices (or grid nodes). The mean values of the fluxes are then obtained by averaging along the celledges.1.6.1 Weighted Cell Residual ApproachFor the description of the weighted cell-residual approach the laminar Navier-Stokes equations are consideredin their conservative formulation:∂w∂t + ∇ · ⃗F = 0, (1.107)where w is the vector of conserved variables and ⃗ F is the corresponding flux tensor. For convenience,the latter is divided into an inviscid and a viscous part, ⃗ F = ⃗ F I (w) + ⃗ F V (w, ⃗ ∇w). The spatial termsof the equations are then approximated in each control volume Ω j to give the residualR Ωj = 1V Ωj∫where ∂Ω j denotes the boundary of Ω j with normal ⃗n.∂Ω j⃗ F · ⃗n dS, (1.108)This cell-vertex approximation is readily applicable to arbitrary cell types and is hence straightforwardto apply for hybrid grids. The residual (1.108) is first computed for each element by making use of asimple integration rule applied to the faces. For triangular faces, a straightforward mid-point rule isused, which is equivalent to the assumption that the individual components of the flux vary linearly onthese faces. For quadrilateral faces, where the nodes may not be co-planar, in order to ensure that theintegration is exact for arbitrary elements if the flux functions do indeed vary linearly, each face is dividedinto triangles and then integrated over the individual triangles. The flux value is then obtained from theaverage of four triangles (two divisions along the two diagonals). This so-called ’linear preservationproperty’ plays an important part in the algorithm for ensuring that accuracy is not lost on irregularmeshes. Computationally, it is useful to write the discrete integration of (1.108) over an arbitrary cell asR Ωj = 1N d V Ωj∑i∈Ω j⃗ Fi · ⃗ dS i , (1.109)60

Cell-Vertex Discretizationwhere F ⃗ i is an approximation of F ⃗ at the nodes, N d represents the number of space dimensions and{i ∈ Ω j } are the vertices of the cell. In this formulation the geometrical information has been factoredinto terms dS ⃗ i that are associated with individual nodes of the cell but not faces; dSi ⃗ is merely theaverage of the area-weighted normals for triangulated faces with a common node i, i ∈ Ω j . Note, thatfor consistency one has ∑ i∈Ω dSi ⃗j= ⃗0. A linear preserving approximation of the divergence operatoris obtained if the volume V Ωj is defined consistently assince ∇ · ⃗x = N d .V Ωj = 1N 2 d∑⃗x i · dS ⃗ i , (1.110)i∈Ω jOnce the cell residuals are calculated, one may then define the semi-discrete schemedw kdt= − 1 V k∑j|k∈Ω jD k Ω jV Ωj R Ωj , (1.111)where DΩ k jis a distribution matrix that weights the cell residual from cell center Ω j to node k (”scatteroperation”),∑and V k is a ‘control volume’ associated with each node. Conservation is guaranteed ifk∈Ω jDΩ k j= I. In the present context, (1.111) is solved to obtain the steady-state solution usingexplicit Euler or Runge-Kutta time–stepping.The family of schemes of interest makes use of the following definition of the distribution matrix:D k Ω j= 1n n(I + C δt Ω jV Ωj⃗ AΩj · ⃗ dS k ), (1.112)where n n is the number of nodes of Ω j and ⃗ A is the Jacobian of the flux tensor. The simplest ‘centraldifference’ scheme is obtained by choosing C = 0 and is neutrally stable when combined with Runge-Kutta time-stepping. A Lax-Wendroff type scheme may also be formulated in which case C is chosento be a constant that depends on the number of space dimensions and the type of cells used — itmay be shown that this takes the simple form C = n 2 v/2N d .61

DESCRIPTION OF THE NUMERICAL TOOL62

Part IIFlame Transfer Functions

Table of Contents2 Linear models for FTFs (Flame Transfer Functions) 692.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2 Models for FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.2.1 Laminar FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.2.2 Turbulent FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Convective τ hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Global formulation for FTF model . . . . . . . . . . . . . . . . . . . . . . . . . 75Local formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Normalizations of FTF amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 77Pressure/velocity FTF model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783 Measurement methods for FTF in LES 813.1 LES of FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.1.1 LES as the proper tool for FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.1.2 Introducing acoustic waves through the burner inlet . . . . . . . . . . . . . . . . 813.2 Postprocessing methods for local FTF . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2.1 ”Time delay” or ”Flight time” method (TD method). . . . . . . . . . . . . . . . 833.2.2 Harmonic forcing; FFT postprocessed method (HF-FFT method). . . . . . . . . 833.2.3 White noise forcing; FFT postprocessed method (WN-FFT method). . . . . . . . 843.2.4 White noise forcing; Wiener-Hopf postprocessed method (WN-WH method). . . 843.3 Preliminary comparisons of postprocessing methods. . . . . . . . . . . . . . . . . . . . 853.3.1 Validation of signal post-processing methods: FFT versus Wiener-Hopf inversion. 86First test : influence of a noisy response signal . . . . . . . . . . . . . . . . . . 86Second test : influence of signal sampling . . . . . . . . . . . . . . . . . . . . . 87Third test : influence of a shift of the response frequency. . . . . . . . . . . . . . 87

TABLE OF CONTENTS3.3.2 Admittance of a diffuser : HF-FFT versus WN-WH methods. . . . . . . . . . . 89Mean flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Admittance calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 Transfer functions of flames 954.1 Configuration A : laminar planar premixed flame (1D) . . . . . . . . . . . . . . . . . . 974.1.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1.2 Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.1.3 Time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.2 Configuration B : laminar V flame (2D axi-symmetric) . . . . . . . . . . . . . . . . . . 1014.2.1 Comparison of global FTF (F norm ) . . . . . . . . . . . . . . . . . . . . . . . . 101Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.3 Configuration C : turbulent burner in cylindrical chamber (3D) . . . . . . . . . . . . . . 1044.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.3.2 A double-swirler partially premixed burner . . . . . . . . . . . . . . . . . . . . 1074.3.3 Large Eddy Simulations for gas turbines . . . . . . . . . . . . . . . . . . . . . . 1074.3.4 Introducing acoustic waves through the burner inlet . . . . . . . . . . . . . . . . 1094.3.5 Reacting steady flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.3.6 Forced reacting flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.7 Coherent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.3.8 Comparison of LES, RANS and experiment : global transfer function . . . . . . 1184.3.9 Comparison of LES, RANS and experiment : local transfer function . . . . . . 1184.3.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4 Configuration D : turbulent burner in a 15 ◦ sector (3D) . . . . . . . . . . . . . . . . . . 1274.4.1 General computational remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.2 Reference signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Comparison of global FTF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Comparison of local FTF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.3 Comparison of global FTF (F norm ) . . . . . . . . . . . . . . . . . . . . . . . . 130Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.4.4 Local comparison of FTF (F dim ) . . . . . . . . . . . . . . . . . . . . . . . . . . 132Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13266

TABLE OF CONTENTSTime delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Weighted Probability density functions of time delays. . . . . . . . . . . . . . . 1354.4.5 Concluding remarks for configuration D . . . . . . . . . . . . . . . . . . . . . . 1374.5 Evaluation of FTF measurements methods in LES. . . . . . . . . . . . . . . . . . . . . 13867

TABLE OF CONTENTS68

Chapter 2Linear models for FTFs (Flame TransferFunctions)2.1 Historical backgroundSince the beginning of the 50’s, the development of higher power propulsion systems has been disruptedby the issue of thermoacoustic instabilities. Thanks to the use of mathematical tools from the field ofelectronics, like the notion of transfer functions, the understanding of these instabilities has also grownat the same time. There are two main methods to study the stability of combustors:• the Purely Acoustic (PA) method. In this approach, the burner is considered as a black box and atwo-port formulation is used to reconstruct a transfer matrix linking acoustic fluctuations on bothsides of the burner [1, 95, 98, 99, 111]. Figure 2.1 illustrates the idea that the whole combustionchamber can be treated as a black box and its acoustic response obtained from the four signals ofinlet and outlet velocity and pressure fluctuation signals. Uinlet ′ and U outlet ′ are respectively theFigure 2.1 - The combustion chamber seen with a two-port formulation.velocity fluctuations at the inlet and at the outlet of the configuration. Pinlet ′ and P outlet ′ are thepressure fluctuations at the inlet and at the outlet of the system. These quantities are linked by the

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)acoustic matrix M of the configuration by the following relation :( ) ( )U′outletU′Poutlet′ = M inletPinlet′(2.1)Note that the coefficients of the M matrix are complex. The element containing the flame is treatedlike all others so that the flame effect is included in the matrix of this element. The major drawbackof this method is that in experiments, the measurement of unsteady pressure and velocities in burntgases can be a difficult task.• the Flame Transfer Function methods. With this method, the velocity and pressure signals after theburner are not used anymore. The transfer acoustic matrix is recovered using the inlet velocity (andpressure) signal(s) as well as the instantaneous heat release signal. When using these methods, thedimensional flame transfer function is called F dim (ω) = Ω′uand a priori depends on the frequency.′|F dim (ω)| is amplitude of the response and Arg(F dim (ω))/(2πω) is the time delay of the response.Depending on the model used, both can vary with frequency or not.As shown by Truffin et Al.[150], both methods (PA and FTF) are mathematically equivalent and a twoportmatrix can be obtained by the knowledge of the transfer function F dim (ω).In the following, focus is mainly put on Flame Transfer Function models and measurement methods.Figure 2.2 illustrates the idea that in a combustion chamber, the whole reaction zone can be treated asa ”black box” and that the global heat release response of the flame can be correlated to the incomingfluctuating velocity and/or pressure signal(s). The first introduction of flame transfer function is due toFigure 2.2 - FTF representation of the reaction zone.the work on linear stability of combustor of Gunder et al. [49] for whom the flame is seen as a reactingdevice with a fixed time delay to mass flow fluctuations coming from the inlet. Following this workand assuming that heat release fluctuations are mostly due to the inlet velocity fluctuations 1 ; Crocco etal. [31] introduced and developed in 1951 the idea that the heat release fluctuation is linked to the inletvelocity fluctuation by a simple relation known as the ”n−τ” model. The n−τ model is the simplest FTFapproach. It assumes the following relation between the heat release fluctuation in the whole chamberΩ ′ Tand the inlet velocity fluctuation:∫Ω ′ T = ˙ω T ′ (t)dv ∝ nu ′ (t − τ) (2.2)V1 This assumption comes from experiments where the flame surface varies linearly with the velocity in front of the flame.70

2.1 Historical backgroundwhere ˙ω ′ T is the local heat release fluctuation and u′ the reference (inlet) velocity fluctuation. n and τare constant over time and do not depend on the frequency of the fluctuation. This model implies thatany inlet velocity fluctuation ”acts” on the flame after a time delay τ under the shape of a heat releasefluctuation whose amplitude is proportional to n times the initial velocity perturbation. The previousrelation assumes that the incoming perturbation is harmonic. It is intrinsically linked to the linear stabilitystudies of combustion systems.In these first studies, n and τ are constant. This model therefore has the drawback of considering allfrequencies equivalent concerning the flame transfer function. But it is known for a long time (De Soetein 1964 [142] for laminar flames and Becker et al. in 1971 for turbulent flames [6]) that FTF are notconstant and can be approximated in amplitude by a frequency low-pass filter. Figure 2.3 representsFigure 2.3 - Typical shape of the flame transfer function. The flame behaves as a low pass filter with a givencut-off frequencya typical amplitude of the FTF as observed experimentally. More recently, Le Helley [52] and Noirayet al. [94] also confirmed this fact experimentally showing that the amplitude of the flame responsedecreases with the frequency.The n − τ model has been extended and refined in many aspects and present FTF models use varyingmodulus and phase with frequency. Variables n and τ are linked to the FTF by the following relations :n(ω) = |F dim (ω)| (2.3)τ(ω) = Arg(F dim(ω))(2.4)2πωYet, the dependance of the time delay (τ(ω)) to the frequency ( ω 2π) is not clear. Other authors [14, 67, 66]have measured almost linearly increasing phase of the flame response with frequency or, in other words,almost constant time delays of the response of the flame. This observation leads to an hybrid modelwhere the modulus of F varies with the frequency but the time delay (τ) is the mean convective timeneeded by the perturbation to go from the inlet of the combustion chamber to the flame. As shown inSection 2.2, this leads to a particular method to measure the phase response of the flame : the ”Timedelay” or ”Flight time” method (TD).71

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)2.2 Models for FTFThe response of turbulent flames to acoustic forcing is studied in the present work. This section focuseson hypothesis and phenomenological models that have been developed for such flows. Great progresshas recently been made in the comprehension of the linear and non-linear response of laminar flames toacoustic disturbances [42, 74, 134]. The derived models will be mentioned since they illustrate the workin progress in the derivation of models for FTF of turbulent flames. First, the model of Ducruix et al. [42]is discussed as a particular example of a theoretical laminar FTF model and its main trends are analyzed(Section 2.2.1). Then phenomenological models for turbulent FTF are discussed along with their mainhypothesis (Section 2.2.2).2.2.1 Laminar FTFThe laminar flame transfer function model proposed by EM2C in a series of publications [42, 94, 134]is dedicated to laminar flames anchored as V flames. The flame movement is studied analytically orusing the so-called G-equation. The fluctuation of the flame surface is correlated to the fluctuation of theincoming velocity and the fluctuation of heat release is directly linked with the fluctuation of the flamesurface. Ducruix et al. [42] and Schuller et al. [134] develop the following theoretical model for theresponse of laminar conical-flame :Ω ′ TΩ ¯= ā 2T u ω⋆2 [(1 − cosω ⋆ )cos(ωt) + (ω ⋆ − sinω ⋆ )sin(ωt)] (2.5)Ω ′ TΩ ¯= u′T ū F norm(ω) (2.6)where ω ⋆ ≈ ωL f α 0 /S l is a reduced pulsation. L f , S l and α 0 are respectively the flame length, thelaminar flame velocity and the half angle of the cone defined by the flame front (see fig 2.4). Ω ′ T and Ω ¯ Tare respectively the fluctuation and the mean value of the volumic integrated heat release. u ′ = acos(ωt)and ū are the fluctuation and the mean value of the velocity field. Equation 2.6 leads to the followingvalues for |F norm (ω)| and Arg(F norm (ω)):|F norm (ω)| = 2/ω 2 ⋆ ∗ [(1 − cosω ⋆ ) 2 + (ω ⋆ − sinω ⋆ ) 2 ] 1 2Arg(F norm (ω)) = tan −1 [(ω ⋆ − sinω ⋆ )/1 − cosω ⋆ )]Figure 2.5 shows a sketch of the results obtained both theoretically and experimentally. It shows a goodagreement between experimental results and model both for the amplitude and the phase of the FTF forlow reduced pulsations ω ⋆ . These results point out some of the main features of the FTF to be met withturbulent FTF.• The FTF can be well reproduced using a first-order low-pass filter at least for low frequencies. Athigher frequencies, the heat release fluctuation and the velocity fluctuation at the burner mouth canbe out of phase whereas the maximum phase is π/2 for a first order low-pass filter. Obviously, theFTF phase in turbulent cases will take any value between 0 and 2π making it hard to represent it72

2.2 Models for FTFFigure 2.4 - Theroretical experiment used for the derivation of the laminar FTF model. a) steady state. b) forcedcase ([42])Figure 2.5 - Sketch of the amplitude and phase of the laminar FTF. Experiment First-order model([42])73

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)with low-order low-pass filter models which are limited to less than 2π for the maximum phase ofthe FTF.• Even with no reflections, the FTF amplitude is not continuously decreasing with the frequency andshows peaks. One might infer that if any reflection on walls should occur, the flame might supportinstabilities at the proper frequencies.Yet, this laminar FTF theoretical model cannot be directly applied to turbulent combustion because ofmultiple hypothesis that cannot be used with turbulent combustion.• Velocity fluctuations are directly correlated to pressure oscillations. In other words, the baselineflow is laminar and no turbulent uncorrelated velocity fluctuations are present.• The speed of the flame front is assumed to be independent of the local curvature and is equal tothe laminar flame velocity.• The flame front is continuous. No local extinction occur, which will be the case for turbulentflames.• The flame is anchored to the burner lip. No movement of the flame base is considered.• The incoming acoustic perturbation is one-dimensional which is not the case for turbulent combustionwhere the flow field is greatly perturbed by the flame.If one of the previous hypothesis is relaxed, the equations that lead to the laminar FTF become toocomplex to be solved by hand and need to be solved numerically. Other models of FTF for laminarflames have been described by Hathout et al [51] or Lieuwen [73] but they have similar limitations.2.2.2 Turbulent FTFMethods used for laminar FTF cannot be directly applied to FTF of turbulent flames. One therefore needsto consider more phenomenological hypothesis along with numerical tools to obtain the turbulent FTF.This section focuses on models for turbulent Flame Transfer Function. These models are all extentions ofthe classical n − τ model [31] and therefore suppose that the response to a harmonic forcing of the inletvelocity is a harmonic heat release fluctuation at the same frequency. Still, it is of interest to mentionthe work of Armitage et al. [5] who recently investigated the non-linear response of turbulent premixedflames.Convective τ hypothesisAs presented in the introduction, most qualitative analysis of the instability phenomenon often mentionthat it is enough to know the time delay since this parameter will impose the frequency of the instability.Therefore most efforts have been made to determine the phase of the flame transfer function and not the74

2.2 Models for FTFgain. Modeling the time delay of the flame response as being constant is something widely done whenconstructors want to have a rough idea of the stability of their combustor [77].The assumptions used in this case are:• that the upstream generated perturbation has a convective nature and therefore that the flame respondsto a propagation of an entropy, a vorticity or a mass flow rate fluctuation which is convectedby the mean flow,• depending on the authors, different distributions of time delays between heat release fluctuationsand forcing velocity fluctuations are chosen, from continuous [39] to discrete [66] distributions. Atleast a distribution for τ should be chosen for partially premixed flames as argued by Dowling [39]who gives a simple explanation in the case of lean combustors. At lean conditions, the equivalenceratio fluctuation in front of the flame is the main reason for heat release fluctuation. A simple flametransfer function is in this case :Ω ′ TΩ ¯= k φ′(2.7)T¯φIf there is a uniform convection time τ from the fuel injection to the flame, thenφ ′ (t)¯φ = −u′ (t − τ)ū(2.8)where u is the fuel injection velocity. But substituting relation 2.8 into relation 2.7 leads to a FTFamplitude which is independent of the frequency; which cannot be true, as argued in the previoussection. Choosing an homogeneous repartition of the convective time between τ − δτ and τ + δτleads to the following relation between heat release fluctuations and inlet velocity fluctuations.ˆ Ω ′ T (ω)¯ Ω T= −k û(ω)ū12δτ∫ τ+δττ−δτe −iωt = −k sin(ωδτ) û(ω)ωδτ ū e−iωτ (2.9)The shape of the FTF obtained from this model better fits the experimental and numerical measurementsof Flame Transfer Function amplitudes which state that the FTF amplitude decreaseswith the frequency of forcing.Global formulation for FTF modelA normalized version of this model is often used to compare the global heat release in the burner and theinlet velocity. It enables to compare results from one configuration to an other. In this case, the transferfunction F is defined by:γ − 1˙ω Tγp 0 S inj∫V′ (t)dv = F (ω)u ′ (t) (2.10)where S inj is the forced reference inlet surface, ˙ω ′ T and u′ are respectively the local heat release andthe inlet velocity fluctuations taken at the reference point. This normalization is introduced by analyticalcalculation of the asymptotic behavior for infinitely low frequency flame response as derived in [61] andpresented below : for premixed flames, this asymptotic value links the ratio of burnt gas temperature (T b )75

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)and fresh gas temperature (T f ) with the amplitude of the flame response to infinitively low frequencyforcing.Replacing integral heat release by fuel consumption in Eq.(2.10) and balancing fuel consumption withfuel supply leads to :∫QS inj u inj ρ 0 Y F = ˙ω T dv (2.11)where Q is the heat released by the combustion of 1kg of fuel. Then in the low frequency limit (ω → 0),an estimate can be derived for the limit value of the amplitude of the FTF (|F (ω)|) :V|F (ω)|(ω → 0) = γ − 1γp 0Qρ 0 Y 0 F min(1, 1/φ) (2.12)Assuming constant heat capacity (c p (T b − T f ) = QYF 0 min(1, 1/φ)), the previous equation can be written:lim |F (ω)| = T b− 1 (2.13)ω→0 T fwhich can be a useful upper limit for F (ω). For typical methane flames, it lies between 1 and 10,depending on the inlet temperature.Local formulationConventional FTF model focus on the correlation between the volumic integral of heat release fluctuationswith a reference point velocity fluctuation. It implies that the flame is compact compared to thecharacteristic wavelength of the perturbation. But although the flame is acoustically compact, it is oftennot compact compared to the velocity fluctuation characteristic lengths. For such cases, an extension ofthese global models can be obtained by linking the correlation of the local heat release fluctuation withthe reference velocity fluctuation. The objective of this extension of the FTF model is therefore to givethe spatial distribution of the Flame Transfer Function. Experiments can also be used to provide thesame information by measuring unsteady heat release in 1-D sections (using a moveable slit (Varoquieet al [152])) or in 3-D using local diagnostics (Poinsot et al [103]).This extension can give either 1-D or3-D representations of the FTF :• For the 3D extension, the relation is :γ − 1γ ¯pS injˆ˙ ω ′ T (x, y, z) = F (ω, x, y, z)û′ (2.14)where ω ˆ˙T ′ (x, y, z) is the local fourier transform of the heat release fluctuation. This model isdirectly linked with the usual FTF model by :∫|F (ω)| =∣ F (ω, x, y, z)dv∣ (2.15)v(∫)Arg(F (ω)) = Arg F (ω, x, y, z)dv(2.16)76v

2.2 Models for FTF• For the 1D extension, the relation is :γ − 1Ω′ˆγ ¯pS T(x) = F (ω, x)û′ (2.17)injwhere Ω ˆ′T(x) is now the Fourier transform of the integrated (over y,z) heat release fluctuation. Thismodel is directly linked with the usual FTF model by :∫|F (ω)| =∣ F (ω, x)dx∣ (2.18)x(∫ )Arg(F (ω)) = Arg F (ω, x)dx(2.19)xThis model extension gives 1D curves of the flame transfer function which can be used in 1Dacoustic codes to gives eigen modes of reacting configurations where the flame cannot be consideredas compact anymore (for example in the ICLEAC experiment of Varoquié [151] or Truffin[149].)Normalizations of FTF amplitudesTwo other normalizations both for the global and local formulations of the FTF are used in this work.The dimensional FTF amplitude is also used.• The FTF can be normalized by the ratio between the mean velocity at the reference point andthe integral of the mean heat release over the whole volume leading to a non-dimensional FTFamplitude. In other words :F norm (ω) = ū¯ Ω TˆΩ′ Tû ′ (2.20)This normalization is used for the analysis of global FTF in sections 4.2 and 4.4.• For the local formulation, the FTF amplitude can also be normalized by its maximum value in thecombustor. This normalization is introduced in section 4.1 for the analysis of the main features ofthe response of 1D flames to inlet velocity forcing. It is described by the following relation:|F norm2 (ω)(x, y, z)| =∣∣ ˆω ′ Tû′∣∣max x,y,z( ˆω′ Tû′)∣ ∣∣(2.21)• Finally, the dimensional amplitude FTF is also used in section 4.3 and 4.4 for the local analysis ofthe flame response. The following relation links the amplitude of the FTF in this case with heatrelease and velocity fluctuations:|F dim (x, y, z)| =∣ˆω ′ Tû ′ ∣ ∣∣∣∣(2.22)77

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)Note that all the ( formulations ) for the amplitude of the FTF ( do)not influence the FTF phase which remainsequal to ArgˆΩ′ Tûfor the global analysis and to Argˆω′ T′ ûfor the local one.′Pressure/velocity FTF modelAll the previously exposed extensions for the FTF model heavily rely on the choice of the reference point.Truffin and Poinsot [150] have shown that the distance between the reference point and the flame zone hasto be compact compared to the acoustic wave length to give relevant amplitude and phase values whenonly the velocity is measured at the reference point. These values can then be used to reconstruct theacoustic matrix of the combustor. This can be an issue for experimentalists since measuring a velocityfield near the flame front requires expensive heat resistant probes. Truffin and Poinsot show that thisissue can be avoided by measuring also pressure fluctuation at the reference point. In this case, theglobal model has four parameters :∫ωˆ˙T ′ dv = F u (ω)û′ + F p (ω) ˆp ′ (2.23)Vwhere F u , F p respectively represent the transfer function of the response due to the velocity and pressurefluctuations. When ∫ ˆ˙VωT ′ is expressed as a function of û′ and ˆp ′ at the reference point, this referencepoint can be anywhere and the difficulty of the classical model (based on ∫ ˆ˙VωT ′ versus û′ ) disappears.However there is a drawback to this approach : to determine F u and F p , two independent calculationsmust be performed or two different forced experimental cases must be set : one forced at the inlet andthe other one forced at the outlet. Truffin and Poinsot show that in this case, the flame transfer functioncan be used to reconstruct the global acoustic matrix of the combustion chamber linking pressure andvelocity fluctuations upstream and downstream of the flame. To assess this achievement, in [150] theacoustic matrix is reconstructed with two methods.• The first method uses the pressure and velocities upstream and downstream of the flame front. Theacoustic matrix is recovered using usual acoustic relations.• The second method uses only pressure and velocity measurements upstream from the flame. Finallythe FTF obtained from their model is used to reconstruct the acoustic matrix of the combustionchamber.The two methods give similar results and these results do not depend on the distance between the referencepoint and the flame front.General remarksAll previously presented formulations (except the pressure/velocity FTF model) rely on the compacityof the flame front compared to the acoustic wave length. If the flame is not compact, the obtained FTF isnot compatible with the PA approach. This means that there is a lower limit to the acoustic wavelength78

2.2 Models for FTFthat can be handled properly by these models. For actual combustion chambers used in gas turbines, thecharacteristic length of the flame is a few decimeters. This means that those models can be applied foracoustic waves which have at least a wavelength (L) of a few meters. In hot gases (c = 900m.s −1 ), thefrequency cut-off (f off ) is therefore of a few hundred Hz only (f off = c L). Using other models than the”pressure/velocity FTF model”, the system identification of combustion chambers is therefore limited tofrequencies lower than a few hundred Hz.79

LINEAR MODELS FOR FTFS (FLAME TRANSFER FUNCTIONS)80

Chapter 3Measurement methods for FTF in LES3.1 LES of FTFThis section describes how simulation can be used to measure FTF in flames (laminar or turbulent).3.1.1 LES as the proper tool for FTFConsidering the measurement of turbulent FTF, the advantage of LES is clear. Turbulent FTF are mainlyneeded for acoustic study purposes in complex geometries. Typical acoustic frequencies are of the orderof 500 Hz in gas turbines corresponding to a period of 2 ms which correspond to time scales that arenow affordable for LES even in complex geometries thanks to the use of unstructured meshes.To determine FTF, one needs either the instantaneous volumic integral of the heat release fluctuation(to determine the global response of the flame) or the instantaneous field of heat release fluctuations(to determine the local response of the flame). In this work, LES (AVBP code from CERFACS(www.cerfacs.fr/cfd)) is chosen to get these fields.3.1.2 Introducing acoustic waves through the burner inletTo determine the transfer function of a burner, the usual procedure is to introduce an acoustic wave intothe burner (usually through the inlet) and measure the perturbation of heat release. The phase betweenthe incoming unsteady flow rate and the unsteady heat release is an essential ingredient of acousticapproaches for combustor stability [32, 55, 69, 96, 107, 113]. However, the exact numerical procedureto introduce acoustic waves in a computation is a difficult topic and may lead to numerical artefacts[41, 105]. In subsonic compressible LES, each boundary must be specified in terms of mean conditions(velocity or pressure for example) but also in terms of acoustic impedance. Many methods for such

MEASUREMENT METHODS FOR FTF IN LESproblems are based on characteristic methods [48, 107] but must be used with care: the well - known‘non reflecting’ conditions actually impose an impedance (which is often unknown) and can have astrong influence on the results [138]. For a simple laminar flame, Kaufmann et al [61] show that a propermethod to excite a combustion chamber is to pulsate the incoming acoustic wave and not the incominglocal velocity. This procedure is used here and avoids false numerical resonances when the chamber isforced. The acoustic wave is therefore introduced into the combustion chamber through the inlet. Notethat when this forcing technique is used, the velocity evolution at the inlet is not fixed anymore anddepends on the combination of the imposed wave (A + ) entering the domain and the one coming back(A − ) from the burner. The inlet velocity and pressure are linked to the acoustic waves at the inlet thoughthe following relation:u ′ = 1 ρc (A + − A − ) (3.1)p ′ = (A + + A − ) (3.2)That means that the inlet velocity fluctuation can only be known before the run if the acoustic wavecoming from the combustor is negligible.Another issue when forcing such a configuration comes from the amplitude of the waves reflected at thechamber outlet which must remain low in order to minimise self-excited oscillations. This is done usingthe NSCBC boundary method [86, 105, 107] which allows the control of the acoustic impedances onthe boundaries.3.2 Postprocessing methods for local FTFThis section presents four methods related to the identification of turbulent combustors. These fourmethods are expected to give the local FTF. Table 3.1 summarizes their characteristics. Sections 3.2.1 to3.2.4 present their principles.Method Inlet forcing Inlet data Outlet dataTDHF-FFTWN-FFTWN-WHNoneSinglefrequencyFilteredwhite noiseFilteredwhite noisePositionreferencepointofMean flamepositionPost processingmethodMeasure flight timeu ′ (t) ˙ωT ′ (x, y, z, t) Fast Fourier Transformu ′ (t) ˙ωT ′ (x, y, z, t) Fast Fourier Transformu ′ (t) ˙ωT ′ (x, y, z, t) Wiener-Hopf inversionAdvantageNo forcedcomputationneededPreciseFastFastTable 3.1 - Characteristics of methods for system identification.82

3.2.1 ”Time delay” or ”Flight time” method (TD method).3.2 Postprocessing methods for local FTFThis method only focuses on the determination of the time delay between heat release and inlet velocityforcing fluctuations using a steady state CFD result [14]. As explained in the introduction, the delaybetween these two fluctuations is considered as the main information for the stability of the combustor.This method consists of three main steps:• First a steady state simulation of the configuration is done (using RANS) or the mean field of anunsteady computation is post-processed (using LES or URANS).• The mean position of the flame is extracted from the field of heat release.• Using the stream lines of the mean flow, the convection time between the inlet and the flame isidentified as the time at the intersection of these lines with the mean flame position. This provides aprobability density function of the delay between heat release and inlet velocity fluctuations. Notethat the probability density function is therefore independent of the frequency: the same delay isused for all frequencies.This method is not as straigthforward as it seems: infinite time delays can be obtained in recirculatingzones. Moreover, determining the ”age” of flow particules as they move in the combustor often requiresto add Lagrangian formulations to the code or an additional conservation equation.3.2.2 Harmonic forcing; FFT postprocessed method (HF-FFT method).This method consists in introducing a downstream acoustic wave at a fixed pulsation (ω) as described insection 3.1.2. It assumes that the response to a harmonic perturbation of the flow is a harmonic fluctuationof heat release. The local flame transfer function is then obtained through the following relation:F dim (ω) =where F dim compares the local fluctuation of heat release to the fluctuation of reference velocity, ˆ˙ ω ′ T isthe Fourier transform of the local heat release fluctuation and û′ is the Fourier transform of the forcedinlet velocity.The method to retrieve the FTF can be described as follows.• Harmonically forced Large Eddy Simulations of the configuration are done for a discrete set ofexcitation frequencies.• 3D fields of heat release fluctuation are extracted as well as the reference (usually the inlet) velocityfluctuation.• 3D fields of F dim (ω) are obtained using FFTs of ˙ω ′ T (t) and u′ (t) and constructing F dim (ω)through equation 3.3. F dim (ω) is a local information about the location and the phase of theflame response.83ˆ˙ ω ′ Tû′(3.3)

MEASUREMENT METHODS FOR FTF IN LESThis method requires FFTs of ˙ωT ′ at all points of the combustor during its forced operation. In thepresent work, this is obtained by using typically 100 snapshots with 10 snapshots per period which givesa frequency resolution of 1/10th of the forcing frequency. Section 4.3 will present a comparison betweenthe TD, HF-FFT methods and experimental results. This method is expensive because it must be repeatedfor a representative set of discrete frequencies (typically 10 to 50), leading to many forced LES.3.2.3 White noise forcing; FFT postprocessed method (WN-FFT method).This method consists in introducing a downstream filtered white noise at the inlet and to obtain the FTFusing one LES only. The local flame transfer function is then obtained through relation 3.3. The methodto retrieve the FTF for the concerned configuration can be described as follows.• A filtered white-noise forced Large Eddy Simulation of the configuration is done.• 3D fields of heat release fluctuation are extracted as well as the reference (usually the inlet) velocityfluctuation.• 3D fields of F dim (ω) are obtained using equation 3.3 and FFTs of ˙ω ′ T (t) and u′ (t)This method requires FFTs of ˙ωT ′only one forced LES is required.at all points of the combustor but is much faster than HF-FFT because3.2.4 White noise forcing; Wiener-Hopf postprocessed method (WN-WH method).This method using white-noise forcing and the Wiener-Hopf relation inversion was introduced in FTFmeasurements by Schuermans et al. [131] and used for real devices by many authors [113, 110, 156,130]. To use this method, the inlet has to be forced with a filtered white-noise including all frequenciesbetween 0 and a cut-off frequency. This method takes advantage of the Wiener-Hopf equation linkingthe autocorrelation matrix Γ of the inlet signal (inlet velocity), and the cross-correlation vector c :Γh = c (3.4)where h is the impulse response filter (i.e the filter corresponding to the unit response of the system).However, in the present work, h corresponds to the white-noise response filter.The method to retrieve the FTF(F dim ) for the concerned configuration can be described as follows:• A LES forced with filtered white noise is performed.• The response signal (local heat release) and the reference signal (inlet velocity) are stored.• Knowing the white noise inlet velocity signal (s), the autocorrelation matrix Γ can be determined:Γ i,j = 1 MN∑s l−i s l−j for i, j = 0, ..., L. (M=N−L+1) (3.5)l=L84

3.3 Preliminary comparisons of postprocessing methods.• Using the inlet (velocity)(s) and the response (heat release)(r) signals, the cross-correlation vectorcan be obtained :c i = 1 N∑s l−i r l for i = 0, ..., L. (3.6)Ml=Lso that the equation 3.4 can be inverted to obtain h.• Using relation 3.7, the FTF is obtained from the filter h.L∑h k e −iω∆tk = 1 + F dim (ω) (3.7)k=0where F dim is the dimensional FTF given by the following relation :F dim (ω) =The parameter L is the length of the filter and is critical to the validity of this analysis. ∆tL representsthe time memory of the filter. It means that the biggest time delay τ max available is determined by thevalue of L (τ max = ∆tL). Attention has also to be paid not to choose L too big to avoid numericalproblems during the inversion of the Γ matrix since its size varies like L 2 .The greatest advantage of this method is that it gives full three dimensional fields of the amplitude andthe phase response of the flame. It is also able to give the response parameters at all frequencies between0Hz and the cut-off frequency. This work will focus on the validation of this method against the singlefrequencyforcing method (comparison WN-WH/HF-FFT (Section 4.4)).Another interest of the WN-WH method compared to the HF-FFT method is that it can give the FTF atall frequencies below the cut-off frequency with only one LES.ˆ˙ ω ′ Tû′(3.8)3.3 Preliminary comparisons of postprocessing methods.Preliminary comparisons are made to assess the validity of the post-processing methods for system identificationdescribed in Section 3.2. Two main tests are computed :• The first type of identification procedures is based on FFT of response and reference signals. Thesecond type is based on the inversion of the Wiener-Hopf relation. Therefore, in a first test, theinfluence of the signal properties on the response of the signal post-processing method (FFT/WHinversion) is studied.• Then HF-FFT and WN-WH post-processing methods are compared on the identification of theadmittance of a diffuser.None of these examples include combustion or FTF but they allow the verification of the post-processingmethods.85

MEASUREMENT METHODS FOR FTF IN LES3.3.1 Validation of signal post-processing methods: FFT versus Wiener-Hopf inversion.To test the validity of each signal post-processing method used for the identification, different simpletests cases are done. For all tests, the reference signal is s(t) = cos(ωt) with ω = 2πf and f = 100Hz.The total signal is composed of ten periods with twenty points per period except for the second test wherethe number of samples per period is changed. The response signal (mimicking heat release) is alwayscomposed of a signal equal to r(t) = 2.3cos(ω(t − 0.004)) with ω = 2πf and f = 100Hz, except forthe third test where the frequency of the response signal is shifted. The goal of this section is thereforeto verify in which cases these methods are able to get the known amplitude of the response (i.e. |F dim | =2.3) and its phase (i.e. Arg(F dim ) = 2πωτ = 2.5133 rad).First test : influence of a noisy response signalOther frequencies are progressively added in the response signal to test the capacity of each methodto retrieve the information from a noisier signal. Table 3.2 and 3.3 sum up the differents frequencycomponents of the response signals. Note that signal 5 is very different from the reference signal andthat its component at 100 Hz is not the most important.Table 3.4 sums up the amplitudes and phases obtained with both methods from the different responsesignals. Both methods give good results with an error lower than 5% even for the signal 5. FFT resultsare better, retrieving the exact amplitude and phase with an error lower than one percent.Freq.(Hz) Reference signal : n, τ Signal 1: n, τ Signal 2: n, τ100 2.3, 4.10 −3 2.3, 4.10 −3 2.3, 4.10 −350 0, 0 0, 0 0, 0180 0, 0 0.03, 2.10 −3 0.4, 2.10 −3300 0, 0 0.02, 1.10 −3 0.05, 1.10 −3500 0, 0 0.01, 5.10 −4 0.2, 5.10 −4Table 3.2 - Composition of the different response signals : Part 1Freq.(Hz) Signal 3: n, τ Signal 4: n, τ Signal 5: n, τ100 2.3, 4.10 −3 2.3, 4.10 −3 2.3, 4.10 −350 0, 0 1.5, 2.10 −3 1.5, 2.10 −3180 1.5, 2.10 −3 1.5, 2.10 −3 1.5, 2.10 −3300 0.5, 1.10 −3 0.5, 1.10 −3 0.5, 1.10 −3500 1.0, 5.10 −4 1.0, 5.10 −4 3.0, 5.10 −4Table 3.3 - Composition of the different response signals : Part 286

3.3 Preliminary comparisons of postprocessing methods.Signal W-H inversion method: |F dim |, Arg(F dim ) (rad) FFT method : |F dim |, Arg(F dim ) (rad)Reference Signal 2.229, 2.508 2.304, 2.508Signal 1 2.3, 2.512 2.304, 2.508Signal 2 2.31, 2.515 2.3, 2.507Signal 3 2.34, 2.521 2.285, 2.504Signal 4 2.229, 2.548 2.295, 2.507Signal 5 2.23, 2.549 2.286, 2.505Table 3.4 - Results : amplitude and phase obtained with both methods. Exact results : |F dim | = 2.3,Arg(F dim ) = 2.5133radSecond test : influence of signal samplingFor this test, the discretisation of both source and response signals is lowered progressively from 20points per period to the Shannon criterion limit of two time samples per period. The amplitude of theresponse is still 2.3 and the time delay 4 ms, corresponding to a phase of 2.5133 rad. Table 3.5 sums upthe number of samples per period for all the signals used in this test. This table also show the resultsgiven by both methods for the amplitude and the phase of the response. Table 3.5, Fig.(3.1) a) and b)Reference signal Signal 1 Signal 2 Signal 3Sampling (Points/Period) 20 10 4 5FFT : |F dim |, Arg(F dim ) (rad) 2.304, 2.508 2.3, 2.503 2.29, 2.507 2.302, 2.514WH : |F dim |, Arg(F dim ) (rad) 2.229, 2.508 2.3, 2.513 2.288, 2.498 2.289, 2.519Signal 4 Signal 5 Signal 6 Signal 7Sampling (Points/Period) 3 2.5 2.3 2FFT : |F dim |, Arg(F dim ) (rad) 2.354, 2.514 2.397, 2.496 2.404, 2.481 2.373, 2.454WH : |F dim |, Arg(F dim ) (rad) 2.3, 2.513 2.311, 2.522 2.3, 2.513 2.3, 2.513Table 3.5 - Decreasing sampling for source and response signals : results in amplitude and phase. Exact results :|F dim | = 2.3, Arg(F dim ) = 2.5133radshow that both methods give good results even near or at the Shannon limit. They are robust to thedecrease of the time sampling. Note that for the Wiener-Hopf inversion method, the size of the filter (Lparameter) has been decreased with the time sampling to keep the same time memory of the system (seeSection 3.2). This test shows that these methods can be used even with poorly discretized signals.Third test : influence of a shift of the response frequency.For this test, the frequency of the response signal is shifted progressively from 100 to 120Hz. Thetotal time of the response signal is kept constant and is equal to 0.1s which corresponds to a frequency87

MEASUREMENT METHODS FOR FTF IN LESa) b)Figure 3.1 - a) Amplitude of the FTF versus time sampling of signals (number of points per period)b) Phase (rad) of the FTF versus time sampling of signals (number of points per period):Theory FFT ◦ WH.resolution of 10Hz. Since the reference signal frequency is 100Hz, the FFT is therefore supposed toconsider all frequencies lower than 110Hz as being 100 Hz for the resulting amplitude and phase. Forfrequencies very different from 100 Hz , the amplitude and phase should decrease to 0 for both methods.Table 3.6 gives the frequencies of all the response signals considered. It compares the results obtainedwith both methods in amplitude and phase. Fig.(3.2) shows that both methods show a selectivity of aboutReference signal Signal 1 Signal 2 Signal 3Frequency (Hz) 100 101 103 107FFT 2.304, 2.508 2.27, 2.224 1.968, 1.654 0.87, 0.463WH 2.229, 2.508 2.26, 2.195 2.02, 1.545 1.105, 0.252Signal 4 Signal 5Frequency (Hz) 110 120FFT 0.022, 6.118 0.023, 6.126WH 0.25, 5.627 0.233, 5.587Table 3.6 - Increasing frequency of the response signal : results in amplitude and phase. Exact results :|F dim | = 2.3, Arg(F dim ) = 2.5133rad3Hz with an error lower than 10%. The residual amplitude at 120Hz is 1% of the signal amplitude forthe Fourier coefficient method and 10% for the wiener-Hopf inversion method. This test shows that theFourier method has a better selectivity than the Wiener-Hopf method. This means that for real signals,close frequencies might pollute each other for short signals when the post-processing uses the Wiener-Hopf inversion relation.88

3.3 Preliminary comparisons of postprocessing methods.Figure 3.2 - Amplitude of the FTF versus frequency of the response signal : Exact FFT ◦ WH.3.3.2 Admittance of a diffuser : HF-FFT versus WN-WH methods.This section presents a preliminary validation of the WN-WH and HF-FFT methods against the theoryfor the calculation of the admittance at the outlet of a subsonic diffuser. The admittance of a diffuser iscomputed using AVBP and forcing the outlet section (Fig.3.3). Results are post-processed using eitherHF-FFT or WN-WH. Even though this example is not a flame yet, it corresponds to an important issuein combustion instability studies : the measurement of impedance of diffusers and distributors. And forthis example like for FTF computation, post-processing methods are crucial so that this case constitutesa good accuracy check of the post-processing HF-FFT and WN-WH techniques. An other advantage ofthis case is that a theoretical solution is available : here, the theoretical admittance can be evaluated withNOZZLE, a code which solves the analytical relations of Marble and Candel [81]. Computations havebeen done by M. Myrczik [89] during an internship at CERFACS. NOZZLE is a software written byLamarque [71]. It provides the impedance of the diffuser. The mesh used for this comparison is shownin figure 3.3. Table 3.7 presents the geometry, mesh and boundary conditions of the diffuser.Figure 3.3 - Mesh of the diffuserMean flowThe validation starts by making sure that NOZZLE provides the same values for the steady flow asthe LES code AVBP. In order to be able to compare the results obtained by AVBP and NOZZLE, itis important to remind that NOZZLE uses a quasi-1D model in contrast to AVBP (2D). Therefore, thevalues provided by the stabilized AVBP calculation are averaged for each section (here: averaging along89

MEASUREMENT METHODS FOR FTF IN LESGeometry Value Boundary conditions ValuesL x (m) 0,2 u in ( m s ) 30L y (m) 0,015 p in (bar) 0,9292059n x 101 ρ in ( kg ) 1,07478m 3n y 11 T in (K) 300,0γ 1,399 r (J.kg/K) 288,19In AVBP the walls are assumed to be adiabatic with zero normal velocityTable 3.7 - Characteristics of the diffuser and boundary conditionsy) to finally compare them to the values obtained with NOZZLE. Note that the boundary conditions intable 3.7 are imposed on the averaged values, which are then fed to the NOZZLE code.Figure 3.4 presents the compared mean flow. The values obtained with NOZZLE coincide very well withAVBP.Figure 3.4 - Comparison between AVBP/NOZZLE for the mean flow in diffuser.90

3.3 Preliminary comparisons of postprocessing methods.Admittance calculationAs shown in the previous section, the mean flows in AVBP and NOZZLE are the same. This sectionpresents now the calculation of admittance with three methods:• NOZZLE which is based on an analytical model.• LES computations with AVBP using the HF-FFT method.• LES computations with AVBP using the WN-WH method.Figure 3.5 shows how the comparisons proceed. A major question for this calculation is which boundaryFigure 3.5 - Procedure for the comparison of post-processing techniques.condition to select at the entrance of the diffuser. If the diffuser was choked, the admittance would notdepend on the compressor elements located upstream of the choked section. Unfortunately, most diffusersare not choked and computing the diffuser outlet admittance requires to know its inlet admittance.Here, in a first step, one assumes a zero acoustic velocity at the entrance.91

MEASUREMENT METHODS FOR FTF IN LESThus, several calculations have been performed with the HF-FFT method. Every calculation startsfrom the initial solution described in section 3.3.2.An upstream acoustic wave A − e −iωt = ip 0 e −iωt is injected at t = 0 in the exit plane. Once the flow isestablished (after a few acoustic times), the admittance calculation can be done using u ′ and p ′ in theexit plane at the forcing frequency. 18 frequencies from 0 to 3000 Hz are used for the HF-FFT method.For the WN-WH method a low-pass-filtered white-noise (0 < f < 4000Hz) is injected at the outletof the domain. One computation of 50 ms is done in order to match a frequency resolution of 20Hz.In this case, the WN-WH method is approximately 10 times faster than the HF-FFT method because itrequires only one unsteady computation instead of 18.Fig.(3.6) shows the comparison between the three methods (WN-WH/ HF-FFT/ NOZZLE). Resultsobtained with the HF-FFT method match very well the results obtained with NOZZLE. With the WN-WH method, one finds the peaks around the same frequencies, with a phase shift for higher ones. Inaddition, between 1000 and ∼ 1200 Hz, the results of WN-WH deviate significantly from the analyticalsolution.Figure 3.6 - Comparison WN-WH/HF-FFT/NOZZLE for the diffuser (real and imaginary part of admittance). Thesolution obtained by NOZZLE is the reference solution.As mentioned before in section 3.2.4, the quality of results obtained with the WN-WH method,depends on the filter size L used for the Wiener-Hopf signal identification method. The influence of thedifferent L (190, 170, 130) is examined. Fig.(3.7) shows that the values obtained with L = 130 differthe most from the NOZZLE solutions. The filter appears to be too small and hence triggers only smalldelay times.L = 170 and L = 190 seem to be appropriate quantities. As shown on Fig. 3.7, it is not obvious whichis the best one; because for one range of frequency L = 170 gives better results, but for another rangeL = 190 fits better.92

3.3 Preliminary comparisons of postprocessing methods.The overall conclusion is that WN-WH methods are faster than HF-FFT methods but they will alsorequire more care.93

MEASUREMENT METHODS FOR FTF IN LESFigure 3.7 - Comparison of three WN-WH results for L= 190, 170 and 130 for the diffuser (Top: Real part ofadmittance; Bottom: Imaginary part of admittance)94

Chapter 4Transfer functions of flamesResults presented in this chapter have been obtained in four configurations (see Fig.(4.1)):• Configuration A : a laminar planar premixed flame (1D)• Configuration B : a laminar conical flame (2D axi-symmetric)• Configuration C : a turbulent gas turbine burner installed in a round combustion chamber at EBIKarsruhe [14] (3D)• Configuration D : a turbulent gas turbine burner installed in a sector of an annular gas turbine,representative of a real machine (3D)The 4 methods presented in chapter 3.2 will be applied to some or all of these set-ups:• Time Delay or Flight Time (TD) method• Harmonically forced ; FFT post-processed (HF-FFT) method• White Noise forced ; Wiener-Hopf post-processed (WN-WH) method• White Noise forced ; FFT Post-processed (WN-FFT) methodTable 4.1 summarizes the different tests which were performed. In most cases, two methods were usedto allow comparisons. The presentation is organized by experimental set-ups. For each case, the configurationis described first and results on FTF follow.

TRANSFER FUNCTIONS OF FLAMESFigure 4.1 - Configurations used for FTF tests.Method TD HF-FFT WN-WH WN-FFTConfiguration A XConfiguration B X X XConfiguration C X XConfiguration D X X XTable 4.1 - Summary of methods and configurations.96

4.1 Configuration A : laminar planar premixed flame (1D)4.1 Configuration A : laminar planar premixed flame (1D)4.1.1 DescriptionThe goal of this section is to determine the main features of the 1D flame response to an incomingacoustic fluctuation in terms of amplitude and time delays. A 1D flame is forced with an incomingacoustic wave at 200Hz with an amplitude of 0.5m.s −1 . The reference point for velocity is the domaininlet so that the distance between this point and the mean flame front is 1mm. The mean inlet velocity isequal to the laminar flame speed so that the flame is stable when there is no forcing. The structured meshis composed of 1000 quadrilateral elements and 2002 nodes. Its dimensions are 2.10 −2 m × 2.10 −9 m.Figure 4.2 sums up the numerical experiment and shows the location of three sample points A, B andC. The flame width is 0.5mm and the pulsated flame ”extension” (the zone over which the flame movesduring forcing) is 1mm.Figure 4.2 - numeric experimental setup.Signals of heat release are collected along a line crossing the flame. Figure 4.3 gives the heat releasesignals at points A, B and C (normalized by the maximum value) as well as the reference inlet velocity.It is interesting to note that signals of heat release are not harmonic because the pulsated flame extensionis greater than the flame width. This particular feature will be encountered in 3D turbulent configurationssince the pulsated velocity amplitude at the flame front will be even higher than the present value.Only the HF-FFT post-processing technique is applied to the instantaneous signals of velocity and heatrelease collected along the line crossing the flame to determine the general behavior of the local responseof the flame.4.1.2 AmplitudeFigure 4.3 shows that the maximum instantaneous heat release is the same for all three points. But theamplitude of the response (|F norm2 | defined in Eq.(2.21)) differs between A and B and C, as presented on97

TRANSFER FUNCTIONS OF FLAMESFigure 4.3 - a) Signals of heat release at points A, B and C. b) Reference velocity signaltable 4.2. The response at point C which is in the mean flame is almost negligible because the frequencyof the heat release fluctuating signal at that point is not 200Hz. Actually, the flame passes exactly twice atthat location during a period, so that the response at the mean flame position is mainly at 400Hz. The 200Hz component increases as one approaches from the maximum extension position of the forced flamefront as shown on figure 4.4.Point A Point B Point CNormalized response amplitude 0.462 0.555 0.068Table 4.2 - Normalized amplitude of the response at points A, B and C (HF-FFT)4.1.3 Time delayAt points A and B, the flame has almost the same amplitude of response, but very different time delays.Point B is closer to the forced inlet. One therefore expects that the time delay at that point will be smaller.It is not the case and table 4.3 sums up the time delays for points A and B . Point A has the lowest timedelay, and since this point is farer than B from the inlet, it is interesting to explain this result.In the present case, the pulsated flame zone is compact when compared to the acoustic wavelength which98

4.1 Configuration A : laminar planar premixed flame (1D)Figure 4.4 - Normalized amplitude of the response (|F norm2 |) along the axial line.is 1.7m long. Therefore, the velocity fluctuation at point C (which is at the mean flame position) isalmost in phase with the inlet velocity. When the velocity at the inlet increases, it pushes the flame fromC to A. The flame first reaches that point before the inlet velocity decreases and comes back to C. Aftermore than half a period, the flame reaches B. In this case the flame mean position therefore separatestwo differents zones of time delays with a discontinuity of half the time period. Figure 4.5 represents thevalues of time delays along the axial line crossing the mean flame position. This type of behavior willalso be observed in 3D cases where time delays exhibit abrupt changes at the mean flame position.Point A Point BTime delay (ms) 1.228 3.761Table 4.3 - Time delays at points A, B99

TRANSFER FUNCTIONS OF FLAMESFigure 4.5 - Time delay (ms) of the flame response along the axial line.100

4.2 Configuration B : laminar V flame (2D axi-symmetric)4.2 Configuration B : laminar V flame (2D axi-symmetric)This configuration has been extensively studied experimentally by Le Helley [52] and numerically byKaufmann [61] and Truffin [149]. This burner corresponds to a laminar flame. The structured meshused for this study is axisymmetric and takes into account for a large part of the feeding line for acousticreasons (see Figure 4.6). It consists of 19189 nodes and 9318 quadrilateral elements.Figure 4.6 - Laminar configuration B. Le Helley [52]This configuration is used here to compare FTF obtained using either HF-FFT, WN-FFT and WN-WHmethods (see section 4.2.1). The configuration is forced at five frequencies between 100 and 500 Hz.4.2.1 Comparison of global FTF (F norm )Computations have been made at CERFACS by Patrick Schmitt and post-processing has been doneby the thermodynamic group at Munich University. Only a global comparison will be made for thisconfiguration.AmplitudeFigures 4.7 presents the amplitudes of the FTF (|F norm |) obtained with HF-FFT, WN-FFT and WN-WHmethods. The three methods reasonably agree for all frequencies. Yet, HF-FFT and WN-WH methodmatch very well and the WN-FFT method gives a noisy response which shows undesired peaks. Notethat all methods give values of |F norm | above unity for all frequencies. This means that the relativefluctuation of the integrated heat release is bigger than the relative fluctuation of inlet velocity. Forexample, if the amplitude of the fluctuation of the inlet velocity represents 10% of the average inletvelocity, the fluctuation of the total heat release in the system represents 11% of the mean power releasedby the flame at 100Hz and 16% at 400Hz.Using Eqs.(2.10) and (2.20), the amplitude |F | of the normalized FTF at 100Hz can also be estimated.It is equal to 8.02. This value can be compared to the asymptotic value given by Eq.(2.13). With a meantemperature of 2387K in burnt gases and a temperature of 300K at the inlet, the asymptotic value of theinteraction index for low frequencies is equal to 6.96. The value for |F | at 100Hz therefore overestimates101

TRANSFER FUNCTIONS OF FLAMESby 15% this theoretical amplitude.Note that in this case, the amplitude response is not dependant of the filter size for the Wiener-Hopfinversion and the black curve represents the mean response for L parameter varying from 20 to 40.Figure 4.7 - Amplitude of FTF : ◦ HF-FFT WN-FFT WN-WH.PhaseFigure 4.8 compares the times delays (Arg(F norm )/(2πω)) obtained with the three different methods.It shows that HF-FFT and WN-WH methods give almost the same time delay for all frequencies. TheWN-FFT method gives strongly varying results at least for frequencies lower than 200 Hz.It is interesting to note that for all methods the global time delay of the FTF is not constant over thefrequency range for this configuration. This implies that the heat release fluctuation can not be due onlyto a pure convective phenomenon. With values for the time delay varying from 0.05 ms to 0.3 ms, theheat release signal goes from an almost ”in phase” fluctuation to a phase of more than π/4.102

4.2 Configuration B : laminar V flame (2D axi-symmetric)Figure 4.8 - Time delay of FTF : ◦ HF-FFT WN-FFT WN-WH.103

TRANSFER FUNCTIONS OF FLAMES4.3 Configuration C : turbulent burner in cylindrical chamber (3D)The burner used for this study is a modified version of a Siemens hybrid burner used here at atmosphericpressure (maximum power = 400 kW). The burner (Fig. 4.9) is mounted on a cylindrical combustionchamber (Fig. 4.10). The non-reacting and reacting flows (Fig. 4.11) for this burner installed ina slightly different combustion chamber have been computed in previous studies and successfully comparedto experiments in terms of axial and tangential velocities (mean and RMS) and mean temperaturefields [137, 138]. The mesh used in this study contains 311047 nodes and 1777730 elements.This configuration is used in the present work to compare time delays obtained with the TD and HF-FFTmethods.Figure 4.9 - Burner geometry.This section presents a comparison of time delay of turbulent FTFs obtained with TD and HF-FFTmethods. Both results are successfully compared with experimental results in terms of time delay. Thissection mainly shows that the TD method fails to predict the global delay between heat release andforcing velocity fluctuations. Besides, it also shows that the response of the turbulent flame is highly nonhomogeneous and the HF-FFT method gives access to the spatial variations of the turbulent FTF bothfor the amplitude and phase. It therefore indicates locations where the flame responds the most to theincoming acoustic wave depending on its frequency.This section follows almost exactly the paper published in Journal of Turbulence, 6(21):1-20, 2005.104

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.10 - Burner mounted on a circular combustion chamberFigure 4.11 - Iso-surface of heat release colored by axial velocity.105

TRANSFER FUNCTIONS OF FLAMES4.3.1 IntroductionBeing able to predict the forced response of a combustion device is a mandatory step to design ‘stable’burners, which exhibit as little sensitivity to acoustic waves as possible. Indeed, flame acoustic couplingis a phenomenon which is found in multiple combustors[155, 114, 138, 133]: it creates a considerableindustrial risk if it cannot be avoided at the design stage. However, predicting flame / acoustics couplingis still a challenge [65, 93, 150]. Only sophisticated numerical methods can be used to characteriseburner responses in real geometries while at the design stage. LES (Large Eddy Simulations) is nowthe most precise numerical tool for turbulent flames as shown by numerous recent examples [138, 34,57]. However, the validation of LES as a tool to study acoustics / flame coupling is far from complete.Few comparisons including experiments and simulations can be found in the literature on LES of flametransfer functions in complex configurations. The main reason for this, is that flame / acoustics studiesrequire the computation of turbulent combustion (something that LES can do reasonably well) but alsoof acoustics. Even in codes dedicated to acoustics in flames where turbulent combustion modelling isextremely simplified, the prediction of flame responses to acoustic perturbations is a difficult topic [114,65, 30, 15, 17, 107]. Therefore coupling such acoustic analysis with state-of-the-art LES is still an openfield of investigation and one objective of this study is to investigate this issue.This study shows how experiments and Large Eddy Simulations (LES) can be used in realistic geometriesto provide one parameter controlling flame stability: the flame response to acoustic waves entering theburner. This response is an essential ingredient of most acoustic tools for flame / acoustics codes. It ischaracterised through the transfer function between unsteady heat release in the chamber and oscillatinginlet velocity. Using simultaneously LES and experiments also allows an investigation of the variouscoherent structures found in swirling combustors [56]:• It is well known [50, 10] that non-reacting swirled flows exhibit strong hydrodynamic modes calledPVC (Precessing Vortex Core). These spiral modes induce a rotation of the swirl motion axis andhave often been suspected of generating instabilities when combustion is activated even thoughcertain recent LES [138] actually suggest that the PVCs are sometimes damped when combustionstarts.• When a combustor (swirled or not) enters a self-excited mode or when it is forced at large amplitudes,another vortical structure appears: it has a mushroom shape in two-dimensional burners[106] and a ring shape in swirled axisymmetric burners [15]. This structure is typical of impulsivelystarting jets and appears only at sufficiently large amplitudes of the velocity fluctuations.Determining when these structures are found in real combustors and what their impact can be, is still anopen field of investigation which can be addressed with LES and advanced experimental diagnostics asproposed here. For the present study, the target configuration is a large-scale industrial turbine burner(400 kW) to be as representative as possible of real combustion devices. This partially premixed systemwas studied both experimentally (at Univ. Karlsruhe) and numerically (at CERFACS).The experimentalconfiguration is described in Section 4.3.2. The LES tools and the numerical methodology used tointroduce acoustic waves through boundaries are then presented (Sections 4.3.3, 4.3.4) before describingresults in terms of mean flow (Section 4.3.5, 4.3.6). The coherent structures appearing during forcingare discussed in Section 4.3.7 while a comparison of LES and experimental data is performed in Section106

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)4.3.8 in terms of phase between inlet velocity and global heat release. Finally, a local analysis of flameresponse is described in Section 4.3.9 to compare LES results with other estimates of combustion phasesused in Reynolds Averaged (RANS) codes [65] which are based on a measurement of ’flight times’ alongfluid trajectories between burner inlet and combustion zone.4.3.2 A double-swirler partially premixed burnerThe burner used for this study is a modified version of a single Siemens hybrid burner used here atatmospheric pressure (maximum power = 400 kW). This burner (Fig. 4.9) is mounted on a cylindricalcombustion chamber (Fig. 4.10). The configuration used for experiments is described on Fig. 4.12. Fullypremixed methane / air is injected through two coaxial swirlers (diagonal and axial) at an equivalenceratio of 0.8 for the axial and 0.5 for the diagonal swirler. The flow rates injected in the combustionchamber are respectively 0.018kg.s −1 for the axial and 0.162kg.s −1 for the diagonal swirler. The pulsatingunit is a rotating valve modulating the diagonal mass flow rate up to amplitudes of 30 percent. Theunsteady velocity induced by forcing is measured using a hot wire probe located in the diagonal swirler(Fig. 4.12) while the global unsteady heat release is evaluated through a measurement of the OH* radical.Since 90% of the injected mixture is at a constant equivalence ratio, the OH* signal can be reliablylinked to the unsteady heat release [79]. Figure 4.13 displays phase-locked OH* images obtained usingan ICCD camera and an appropriate OH* filter (λ = 307.8nm). Typical flame shapes are observedwithout forcing or with low-frequency forcing (left), and with high-frequency forcing (right). The flameis quasi-steady as long as the frequency of the excitation remains below 50Hz. Beyond this limit, ringvortices interact with the flame imposing their axis-symmetrical shape to the flame.4.3.3 Large Eddy Simulations for gas turbinesRecent studies [138, 3, 19, 29, 102, 124, 117] have shown that Large Eddy Simulations (LES) are powerfultools to study the dynamics of turbulent flames and especially their response to acoustic forcing.However, the extension of LES methods to complex geometries creates new problems to adapt highprecisionnumerical techniques (required for LES) to arbitrarily complex meshes. Here a parallel LESsolver called AVBP (see www.cerfacs.fr/cfd/) is used to solve the full compressible Navier Stockes equationson hybrid (structured and unstructured) grids with second or third-order spatial and temporal accuracy[29]. Subgrid stresses are described by the WALE model [92]. The subgrid flame / turbulenceinteraction is modelled by the Thickened Flame (TF) model [3, 28]. The validity of the TF model usedhere has been checked in many recent papers [138, 121] and will not be discussed here. For the presentapplication, methane / air combustion is modelled using six species (CH 4 ,O 2 , CO 2 , CO, H 2 O and N 2 )107

TRANSFER FUNCTIONS OF FLAMESFigure 4.12 - Experimental setup.quasi-steady flamering vorticesFigure 4.13 - Instantaneous Phase-locked OH* images of the flame structure.108

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)and two reactions [137]. These reactions are :The forward rate of the first irreversible reaction is:and the rate of the second reaction is:CH 4 + 3/2O 2 => CO + 2H 2 O (4.1)CO + 1/2O 2 CO 2 (4.2)q 1 = AY 0.9CH 4Y 1.1O 2exp(−T 1 a /T )q 2 = B[Y CO Y 0.5O 2− (1/K y ) Y CO2 ] exp(−T 2 a /T )where A = 6.1.10 8 mol.m −3 .s −1 , B = 6.1.10 5 mol.m −3 .s −1 , T 1 a = 17613 K, T 2 a = 6038 K andK y (T ) is the equilibrium constant of reaction (2).The mesh used for the burner of this study contains 1.7 million elements. The characteristic size of themesh in the combustion chamber is 3mm which has to compared to the laminar flame thickness (1.5mm).The simulations were achieved in ’Centre Informatique National de l’Enseignement Superieur’ (CINES)and the average computing time needed is around 10 hours on 64 processors on SGI 03800 for one periodat the forcing frequency of 120Hz.4.3.4 Introducing acoustic waves through the burner inletTo determine the transfer function of a burner, the usual procedure is to introduce an acoustic waveinto the burner (usually through the inlet) and measure the perturbation of heat release. The phasebetween the incoming unsteady flow rate and the unsteady heat release is an essential ingredient ofacoustic approaches for combustor stability [107, 32, 55, 69, 96, 113]. Experimentally, this is usuallydone with loudspeakers forcing the flow upstream or downstream of the combustion chamber. At highchamber powers, however, rotating valves are required as done here. Because of the cost of such devicesand the danger of pulsating large flow rates of premixed gases, performing the same task with LES isan obvious alternative path. However, the exact numerical procedure to introduce acoustic waves in acomputation is a difficult topic and may lead to numerical artefacts [105, 41]. In subsonic compressibleLES, each boundary must be specified in terms of mean conditions (velocity or pressure for example)but also in terms of acoustic impedance. Many methods for such problems are based on characteristicmethods [107, 48] but must be used with care: the well - known ‘non reflecting’ conditions actuallyimpose an impedance (which is often unknown) and can have a strong influence on the results [138]. Fora simple laminar flame, Kaufmann et al [62] show that a proper method to excite a combustion chamber isto pulsate the incoming acoustic wave and not the local velocity. This procedure is used here and avoidsfalse numerical resonances when the chamber is forced. Another issue when forcing such a configurationcomes from the amplitude of the waves reflected at the chamber outlet which must remain low in orderto minimise self-excited oscillations. This is done using the NSCBC boundary method [107, 105, 86]which allows the control of the acoustic impedances on the boundaries.109

TRANSFER FUNCTIONS OF FLAMES4.3.5 Reacting steady flowSince the focus of this section is the forced response of the burner, no experimental results will be providedfor the unforced case. The non-reacting and reacting flow for this burner installed in a slightly differentcombustion chamber have been computed in previous studies and successfully compared to experimentsin terms of axial and tangential velocities (mean and RMS) and mean temperature fields [138, 137]this validation is not repeated here.The reacting case corresponds to a global equivalence ratio of 0.51, air flow rate of 180 g/s, a Reynoldsnumber of 120000 (based on bulk velocity and burner diameter) and power of 277 kW. A 2D snapshotof the axial velocity field (Fig. 4.14) shows a strongly turbulent flow. The swirled motion creates a largerecirculation zone in the centre of the combustion chamber and a circular outer recirculation zone (whichappears as two separates zones on the 2D axial cut). An iso-line of temperature (1100K) reveals the flameposition close to the burner mouth, stabilised both on the inner recirculation zone created by swirl and onthe outer recirculation zone. A temperature isosurface at 1000 K (Fig. 4.15) in an unsteady non-pulsatedcase shows the typical flame shape: the edges of the flame are disturbed but the central zone, near theaxial inlet, is very stable due to the injection of richer premixed gases (equivalence ratio of 0.8) throughthe axial swirler.Figure 4.14 - Example of instantaneous field of axial velocity in the axial plane,(1100K). Unforced flow.iso-line of temperature110

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.15 - Snapshot of temperature isosurface (1000 K) coloured by axial velocity. Unforced flow.111

TRANSFER FUNCTIONS OF FLAMES4.3.6 Forced reacting flowIn the experiment, the combustor was forced at frequencies changing from 10 to 120 Hz while the LESwere performed at 80, 120 and 250 Hz. 1The inlet forcing immediately drives a heat release oscillation. Figure 4.16 compares evolutions ofheat release, diagonal inlet velocity and total mass of fuel in the chamber for the 120Hz case. Theinlet velocity (circles) and the enclosed fuel mass (deltas) oscillate sinusoidally, perfectly correlatedto the inlet acoustic pulsation. Heat release exhibits more nonlinearities, especially between instants”φ = 0” and ”φ = π/2”. The fast decreasing of the heat release occurs when the fresh gases previouslyinjected burn out. This ”main event” during the cycle of excitation has been observed experimentallyfor pulsated and self-excited combustors at the origins of the study of combustion instability. mettredes refs. The flame shape (visualised by an isosurface of temperature) is displayed for four phases ofthe cycle (Fig. 4.17) (φ = 0, π/2, π, 3π/2). The phase φ = 0 corresponds to the minimum valueof the axial velocity whereas φ = π corresponds to the maximum value of the axial velocity. Theflame surface varies through the cycle and exhibits various complex structures especially at φ = 3π/2(Fig. 4.17d) where the ring vortex dislocates into smaller structures. The total heat release oscillatesfrom 0.7 to 1.3 times its steady value (Fig. 4.16) at π of the inlet velocity. The flame goes through phasesof rapid expansion (between instants ”φ = π” and ”φ = 3π/2” on Fig 4.17 for example) during whichring vortices created by the forcing distort the flame front and expand it. Later in the cycle, the flameshrinks very fast (between instants ”φ = 0” and ”φ = π/2” on fig 4.17) as already pointed out on fig 4.16.4.3.7 Coherent structuresLES reveals that two coherent structures determine the flame shape and the structure of the flow duringforcing. Figure 4.18 displays the classical ring vortex structure observed in this configuration. Thesestructures are displayed for four phases of the cycle (Fig. 4.18) (φ = 0, π/2, π, 3π/2). The phase φ = 0corresponds to the minimum value of the axial velocity whereas φ = π corresponds to the maximumvalue of the axial velocity. The excitation creates a large axisymetrical ring which is convected by themean flow and interacts with the flame, increasing the flame surface as it passes through. This ring isvisualised here using an iso-surface of the detection criterion of Hussain and Jeong which is based on thesecond invariant of the velocity tensor [59]. When the values of this criterion is positive, the rotation rateis greater than the deformation rate, which implies the presence of a coherent structure. LES shows thatthis ring is coherent and axisymmetric. At times ”φ = π” and ”φ = 3π/2”, the structure is ring-shapedand well identified in Figure 4.18. This ring structure directly shapes the flame surface leading to themushroom-shaped flame of Figure 4.17 ”φ = π” and ”φ = 3π/2”. At instants ”φ = 0i” and ”φ = π/2”however, the ring becomes unstable and loses coherence. At instant ”φ = π/2” , it almost disappears.A second structure generated by the axial forcing is the precessing vortex core. This instability is typicalof non-reacting swirling flows [80] but was not observed for the unforced reacting case. Visualising an1 At low forcing frequencies, the flame response is quasi-steady. Moreover, LES of low frequency cases become expensiveas characteristic physical times increase. This explains why LES were not performed at frequencies smaller than 80 Hz.112

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.16 - Heat release (squares), inlet normal velocity (circles) and enclosed fuel mass (deltas) fluctuations inpercents of mean values versus phase angle for the 120Hz LES.113

TRANSFER FUNCTIONS OF FLAMESφ = 0 φ = π/2φ = 3π/2φ = πFigure 4.17 - Snapshots of temperature isosurface (1000 K) coloured by the axial velocity for four differentphases of the cycle : f = 120Hz.114

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)iso-surface of the vortex criterion (Fig. 4.19) in an inner cone excluding the large vortex ring of Fig. 4.18reveals a coherent structure rotating around the chamber axis. The frequency of this precessing structureis independent of the pulsation frequency and remains around 75 Hz.115

TRANSFER FUNCTIONS OF FLAMES(a) φ = 0 (b) φ = π/2(d) φ = 3π/2(c) φ = πFigure 4.18 - Snapshots of vortex criterion coloured by the temperature (Hussain [59]) for four different phases ofthe cycle : f = 120Hz.116

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)(a) φ = 0 (b) φ = π/2(d) φ = 3π/2(c) φ = πFigure 4.19 - Snapshots of vortex criterion coloured by the axial velocity visualising the precessing vortex corefor four different phases of the cycle : f = 120Hz.117

TRANSFER FUNCTIONS OF FLAMES4.3.8 Comparison of LES, RANS and experiment : global transfer functionFigure 4.20 compares numerical predictions and experimental measurements of the flame transfer functionusing the n − τ model [107, 32]. This model, commonly used when dealing with reacting systemidentification, is based on a first harmonic analysis. Therefore it does not account for non-linear phenomenawhich could be relevant in case of self-excited flames configurations [133, 75]. This model linksthe Fourier transform of the volume-integrated heat release fluctuation ˙Ω ′ T to the Fourier transform ofthe velocity fluctuation at the inlet u ′ by :ne iwτ = γ − 1S D γp 0ˆ˙Ω ′ Tû′(4.3)where w is the forcing pulsation (w = 2π ∗ f), p 0 is the mean pressure in the combustion chamber andS D the diagonal swirler inlet surface of fuel injection. Note that the phase φ is directly calculated fromτ, by multiplying by −w. As no time dependent variables are available in RANS results, φ is extractedusing the ”flight time” method described in subsection 4.3.9. Experimental results are given by the phasecomparison between the signal given by a photomultiplier (with a OH* frequency filter) which is directlylinked to the heat release, and a hot wire probe signal at the inlet of the combustion chamber. Values ofthe LES phase φ are in good agreement with experimental values (Fig. 4.20). The experimental phasebetween heat release and velocity fluctuations is of the order of −3.5 rad at 120 Hz.No comparison of the amplitudes of the Flame Transfer Function given by the HF-FFT method and theexperiment is provided. The reason is that the amplitude of the FTF given by the HF-FFT method greatlychanges with the acoustic boundary conditions. Since the experimental acoustic boundary conditions areunknown, no trustful amplitude can be obtained with the HF-FFT method.Experiment LES RANSφ (rad) -3.5 -3.2 -1.1Table 4.4 - Comparison of phases (rad) for the 120Hz case.4.3.9 Comparison of LES, RANS and experiment : local transfer functionThe previous subsection has shown differences in terms of global phase between LES, RANS and experiments.To analyse these differences, local transfer function must be studied. The usual n w -τ w modellinks inlet velocity perturbations to global heat release providing one delay τ w (expressed here as a phaseφ w = −τ w ∗ w) and one interaction index n w . In simulations, it is possible, however, to construct a mapof local n w -φ w values linking the inlet velocity perturbation to the local heat release variation to have alocal phase φ w (x, y, z) and a local index n w (x, y, z). These quantities are defined by:n w (x, y, z) = MOD( ˆω ′ Tû ′ ) (4.4)118

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.20 - Comparison of LES, RANS and experiment phases.φ w (x, y, z) = ARG( ˆω ′ Tû ) (4.5)where ˆω ′ T and û′ are the Fourier transforms of the local heat release and of the inlet velocity fluctuationsrespectively. MOD and ARG are respectively the real and imaginary part of the complex Fouriercoefficient of the transfer function at pulsation w.Two methods can be used to evaluate n w and φ w :• direct method (using time-resolved fields of the forced flow): using the definitions of equations 4.4and 4.5, ω ˙ ′ T (t, x, y, z) and u ′ (t) inlet can be used to evaluate φ w (x, y, z) and n w (x, y, z) for eachforcing frequency. This method is possible when LES forced fields are available as done here.• ”flight time” method (using averaged fields of the unforced flow): quite often, φ w is extracted fromaveraged fields, simply by measuring the time needed for one particle to travel from the diagonalswirler inlet to a given location in the combustor. In this case, convection delays to the flame frontare evaluated for example by picking flight time values to the isosurface Y CH4 = 0.004 (where themaximum mean heat release occurs.). This method is common when RANS codes are used andno forced response has been computed. It is based on the calculation of the dispersion of a passivescalar in a stationary field. More detailed information about the approach is given by Krebs et al.in [65]. It can also be tested here using the averaged fields given by LES.To avoid ambiguities due to the mean flow used for the ”flight time” method, the averaged LES fieldwas used and no direct RANS code was run. This ensures that the ”flight time” and the direct methods119

TRANSFER FUNCTIONS OF FLAMESare applied on exactly the same flows.Figure 4.21 shows a longitudinal cut coloured by the amplitude response at 120Hz using the directmethod. Since the area including the mean position of the flame mainly responds at twice the frequencyof excitation, the two flame-shaped like structures show the maximum extension of the flame through acycle of excitation (where the flame only goes once during a period of excitation). The flame in front ofthe axial swirler has the strongest response as shown by the high n w values in this zone (Fig 4.21). Thismay be due to the equivalence ratio difference between the two flames, the richer one producing moreheat release and therefore more heat release fluctuation when excited by the acoustic wave.Figure 4.22 displays the same cut coloured by the phase (φ w (x, y, z) given by equation 4.5) betweenheat release and inlet velocity fluctuations and obtained by the direct method. It appears that the flamedoes not move in a compact way but rather gets stretched by the incoming acoustic excitation.Figure 4.23 displays again the same axial 2D cut coloured by the phase φ w obtained with the ”flighttime” method. It is assumed in this analysis that only convective time lower than two periods ofexcitation are worth being considered (i.e. 16.6ms) [65]. It can be observed that certain regions of themean flame are not reached by the flow within two periods after the start of the excitation. This impliesthat the heat release fluctuation is not linked to the convection of an initial perturbation but is the resultof a complex interaction between acoustics and the flow in the chamber. The resulting structures mayappear in the recirculation zone (located in the centre of the combustion chamber (Fig 4.14)) before anypurely convective phenomenon reaches it.Figure 4.24 shows the isosurface of control (Y CH4 = 0.004) coloured by φ w (x, y, z) obtained with thedirect method. It can be seen that the axi-symmetry of the flow is conserved in terms of combustionphase.Figure 4.25 shows the isosurface of control coloured by φ w (x, y, z) obtained with the ”flight time”method. The phases obtained with this method still have an axisymmetrical repartition. As already donefor the axial 2D cut (Figure 4.23), non-represented phases correspond to convective times greater thantwo periods of excitation.Figure 4.26 compares the PDF of phases and its cumulant measured either by the direct methodnecessarily issued from LES (thin line) or by the ”flight time” method (thick line) on the isosurfaceof control Y CH4 = 0.004. The PDF of phases obtained by LES exhibit a few peaks around -3.5 and0 rad while the ”flight time” method gives a most probable value of φ w around -1.1 rad. This meansthat according to this last method the heat release fluctuation should be in phase with the inlet velocityfluctuation signal which is incompatible with the results of Figure 4.16 or with the experimental data .Figure 4.26 is obtained from unweighed phases but it is reasonable to expect that points wherelarge unsteady heat release takes place, will have the largest effect on the instability and that they shouldaccount for a larger part in the average phase. This means that another method to post-process phases isto weigh them using the local n w index value.120

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.27 compares the two weighted PDF 2 of phases using the following weighted phase:P w (φ 0 ≤ φ w ≤ φ 0 + dφ 0 ) =∑ ndφ01 norm i∑ N1 norm iin the region defined by:n w (x, y, z) > n w(x, y, z) max100n dφ0 is the number of mesh points for which φ 0 ≤ φ w ≤ φ 0 + dφ 0 , N is the total number of mesh pointsin the volume andnorm i = Int( 100 ∗ n(x i, y i , z i )n max)One can observe that the two weighted PDF (Figure 4.27) do not match. Although the ”flight time”method shows a wide maximum between -3.5 and 0 rad, the phases obtained with equation 4.5 havean evident maximum at -3.2. This last most represented value of φ w is in agreement with a global heatrelease fluctuation (i.e.. integrated over the whole volume) out of phase with the inlet velocity fluctuation(Figure 4.16 ). Clearly in this example, the ”flight time” method does not give accurate results.2 Note that this normalisation is usually not available in ”flight time” method, because the local n w values of the flametransfer function are needed.121

TRANSFER FUNCTIONS OF FLAMESFigure 4.21 - Longitudinal cut coloured by response amplitude n w (x, y, z) at 120Hz. (Direct method)122

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.22 - Longitudinal cut coloured by phase φ w (x, y, z) (rad) between heat release and inlet velocityfluctuations (wrapped between -2π and 0). iso-line Y CH4 = 0.004. (Direct method)Figure 4.23 - Longitudinal cut coloured by convection phase for diagonal injection (wrapped between -2π and0). iso-line Y CH4 = 0.004. (Flight time method)123

TRANSFER FUNCTIONS OF FLAMESFigure 4.24 - Isosurface of Y CH4 = 0.004 coloured by phase (rad) between heat release and inlet velocityfluctuations. (Direct method)Figure 4.25 - Isosurface of Y CH4 = 0.004 coloured by convection phase (rad) for diagonal injection. (Flighttime method)124

4.3 Configuration C : turbulent burner in cylindrical chamber (3D)Figure 4.26 - Comparison of the phases’PDF and its cumulant on the isosurface Y CH4 = 0.004.PDF : ”flight time” method. ”direct” method.Cumulant : ”flight time” method . ”direct” method.Figure 4.27 - Comparison of the phases’PDF and its cumulant obtained inside maximum extension flame zone.PDF : ”flight time” method. ”direct” method.Cumulant : ”flight time” method . ”direct” method.125

TRANSFER FUNCTIONS OF FLAMES4.3.10 ConclusionsLarge Eddy Simulations are used to compute the forced response of an acoustically excited swirledburner. The configuration is a 1:1 representation of a prototype gas turbine provided by Siemens and inwhich premixed gases are injected through two complex-geometry swirlers at two different equivalenceratios (0.5 and 0.8). The LES solver uses hybrid meshes and high-order schemes. Combustion/turbulenceinteraction is modelled using the thickened flame model. A two-step scheme for methane/air combustionis used to represent chemistry. A comparison of combustion phases given by LES and measurementsperformed in Karlsruhe yields good agreement. Exciting the main inlet at frequencies varying from 80to 250 Hz, two main flow structures are evidenced by LES:• A toroidal structure which modulates the global heat release by interacting with the flame surface.• A precessing vortex core attached to the centre of the axial swirler and whose frequency is independentof the inlet excitation frequency.Finally, the maps of combustion phase and response amplitude obtained by LES show that the phasebetween heat release and inlet velocity fluctuations cannot be simply related to a convective phase fromthe burner inlet to the flame front as currently done when no LES results are available.More generally, this study confirms the potential of LES to study unsteady turbulent combustion incomplex geometries.126

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)This configuration uses almost the same burner as configuration C but mounted on a sector of an annularchamber (see Figure 4.28). The simulation includes a 15 ◦ sector of the whole combustion chamber.Side faces are enclosed with slip adiabatic rigid walls. The mesh contains 339417 nodes and 1859248tetrahedral elements. Validation concerning mean and RMS results will not be presented here. This sameFigure 4.28 - Burner mounted on an annular combustion chamber.mesh has been used to explore the influence of piloting on combustion stability [140]. By adding twoside burners to this configuration, it is also used by Staffelbach et al. [144] to evaluate the influence ofthe extension of the mesh on the resulting unstable modes of the configuration.Configuration D is used here to compare results of FTF (F dim and F norm ) using HF-FFT, WN-FFT andWN-WH methods.127

TRANSFER FUNCTIONS OF FLAMES4.4.1 General computational remarks.Two main methods using LES to get the FTF are compared. The first method (HF method) requires theinjection of a harmonic signal at the main inlet of the configuration. The frequency of forcing is changedto get the dependance of the FTF to frequency. The configuration is forced at six frequencies between90 and 540 Hz.With an explicit resolution of the Navier-Stokes equations and a time step of the order of 4.0e −7 s,25000 iterations are needed to compute one period if the configuration is forced with a 100 Hz acousticwave. Since the HF-FFT method uses the Fourier transform of signals, the spectral resolution is equalto the inverse of the total computed time. Therefore, to get a resolution of 20Hz, 0.05s have to becomputed, which corresponds to 125000 explicit iterations. On 64 processors on a CRAY XD1, itrepresents 50h of computation. Therefore, no calculations for lower frequencies than 90Hz have beenmade to save computational time.Since WN-WH method is based on the correlation of signals and not on their Fourier transform,theoretically only one period has to be resolved to state for the time delay, even if the third simple testshows a selectivity issue concerning the Wiener-Hopf inversion method (see Section 3.3). Therefore, toenable a simpler comparison between the three methods and to avoid this issue, they are applied on thesame fields of heat release and velocity fluctuations corresponding to a minimum of five periods for allconcerned frequencies.Table 4.5 summarizes the physical simulated times and CPU costs for the turbulent case and forthe three methods. All computations have been done on a CRAY XD1 (120 processors).Forced casesHF-FFT 90 HzHF-FFT 180 HzHF-FFT 270 HzHF-FFT 360 HzHF-FFT 450 HzHF-FFT 540 HzHF-FFT (global)WN-WH or WN-FFT (F cut = 600Hz)Turbulent combustorPhysical time: 55ms Cpu Cost : 54hPhysical time: 44ms Cpu Cost : 43hPhysical time: 33ms Cpu Cost : 32hPhysical time: 22ms Cpu Cost : 21hPhysical time: 22ms Cpu Cost : 21hPhysical time: 22ms Cpu Cost : 21hTotal Cpu Cost : 192hPhysical time: 55ms Cpu Cost : 54hTable 4.5 - Computational costs and simulated times for the different runs of the turbulent configuration.Note that the physical simulated time can be decreased as the frequency increases because the periodof the forcing also decreases. Also note that the frequency resolution is not kept constant. No less thanfive periods have been computed for the harmonically forced cases. The last two lines compare the globalcomputational cost of the two types of methods. Since different computations have to be done for each128

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)frequency when using HF-FFT method, the cost of this method is four times bigger. The goal of thefollowing section is therefore to validate WN-FFT and WN-WH methods in an industrial configuration.4.4.2 Reference signalsFor both methods, investigations of global and local FTF are done. Depending on global or local FTFanalysis, different response signals have to be compared to the reference signal.Comparison of global FTF.In this case, the reference inlet velocity signal is the harmonic or white-noise fluctuating velocity, andthe response signal is the integrated heat release fluctuation over the whole volume. Figure 4.29 presentsthese signals for the turbulent configuration and for the white noise forced case. Inlet velocity and heatrelease signals are normalized by their mean values. There is a clear correlation between these twosignals. One can observe that a maximum of inlet velocity induces a maximum of heat release afterapproximately 2-3 ms .Figure 4.29 - Normalized signals of reference for the white-noise forced case: inlet velocity heatreleaseComparison of local FTF.In this case, 3D fields of heat release fluctuation are used. The transfer function of each grid pointis calculated using HF-FFT, WN-FFT and WN-WH methods. Figure 4.30 shows a 2D longitudinalcut colored with the instantaneous local heat release. The LES are done using the optimal local flamethickening model [125] with 7 points in the flame front. It leads to a thickening factor ranging from129

TRANSFER FUNCTIONS OF FLAMESapproximately 5 to 50 depending on the local refining of the mesh. A point shows the location of thereference inlet point. This local treatment enables amplitude and time delay from FTF 3D fields to becompared using different system identification techniques.Figure 4.30 - 2D longitudinal cut colored by instantaneous heat release. Black point marks the reference inletpoint.4.4.3 Comparison of global FTF (F norm )AmplitudeFigures 4.31 present the amplitudes of the FTF obtained with HF-FFT, WN-FFT and WN-WH methodsfrom signals of Fig.(4.29). It appears that HF-FFT and WN-WH methods give similar results for theamplitude of the turbulent FTF, with a minimum taken from the Wiener-Hopf curve at approximately300Hz. In this case, the WN-FFT results are not in agreement with the other results. Figure 4.31 alsoshows that the result obtained with WN-WH method is sensitive to the size of the filter used to computeauto-correlation matrix and cross-correlation vectors. Depending on the value chosen for the filter lengthL, the response experiences a variation of approximately 10 percent. When approaching the cut-offfrequency (600 Hz) of the low pass filter used to obtain the white noise signal, the discrepancy betweenresults for different L values increases. In contrary to what is observed for the laminar FTF amplitudein chapter 4.2, the amplitude of the FTF is not greater than one for the frequencies of interest. It meansthat the relative fluctuation of heat release is lower than the relative fluctuation of the inlet velocity forall frequencies. Figure 4.31 shows that a maximum of the FTF amplitude is reached around 500Hz. If130

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)the inlet velocity is forced at 10% of its mean value at 500Hz, the heat release will fluctuate with anamplitude of 8% of its mean value.Using Eqs.(2.10) and (2.20), the amplitude |F | at 100Hz can be estimated. It is equal to 1.77. This valuecan be compared to the asymptotic value given by Eq.(2.13). With a mean temperature of 1835K inburnt gases and a temperature of 685K at the inlet, the asymptotic value of the interaction index for lowfrequencies is equal to 1.68. The value for |F | at 100Hz therefore compares well with this theoreticalamplitude.Figure 4.31 - Amplitude of FTF : ◦ HF-FFT method. + WN-FFT method. .... WN-WH method for various valuesof the filter size L.PhaseFigure 4.32 compares the times delays for the configuration D obtained with HF-FFT, WN-FFT and WN-WH methods. As for configuration B, it shows that HF-FFT and WN-WH time delays almost match forall frequencies. WN-FFT method gives almost incoherent time delays that can hardly be compared to theother results. Results show a discontinuity around 300Hz. This artefact is only due to the phase crossingthe 2π limit and going back to zero. Without reference time for the delay, values above 300 Hz are eitherτ or T + τ without changing the phase between the heat release fluctuation and the reference velocityfluctuation. Though, fluctuations (depending on L) of the results of Wiener-Hopf relation’s inversion areimportant near 300 Hz and 600 Hz. The last frequency corresponds to the cut-off frequency and hasalready been pointed out concerning the amplitude analysis. The source of fluctuations around 300 Hz isnot clear but corresponds to the fact that the phase crosses the 2π limit for slightly different frequenciesdepending on the value of the L parameter.131

TRANSFER FUNCTIONS OF FLAMESFigure 4.32 - Time delay of FTF : ◦ HF-FFT method. + WN-FFT method. .... WN-WH method (for various valuesof the filter size L).4.4.4 Local comparison of FTF (F dim )It has been shown in the previous section that the global response of the system is almost identical whenforced with a harmonic signal or with a filtered white noise. It is therefore interesting to focus on thelocal response of the flame. The analysis is done by comparing results obtained with the two methodsHF-FFT and WN-WH at 90 Hz. the WN-WH method gives the FTF for all the frequency range includedin the spectrum of the imposed filtered white noise. Results at 90 Hz are therefore extracted from thetotal resolved spectrum for the WN-WH method. The frequency of 90Hz was chosen because it is thevalue observed experimentally in the full combustor (it is an azimutal mode). Longitudinal cuts coloredby the local amplitude and time delay are used to support this analysis. Finally the weighted probabilitydensity function of time delays for both cases is commented.AmplitudeFigures 4.33 and 4.34 show 2D longitudinal plane cuts colored by the amplitude of the FTF at 90Hz,respectively obtained using the HF-FFT and WN-WH methods. Note that the mean flame position isslightly different whether the flame is forced using HF-FFT method or WN-WH method. This is dueto the fact that the simulation time is not long enough to ensure a perfect convergence of the mean heatrelease field.Figure 4.33 shows that the response amplitude to the 90Hz acoustic excitation is almost axi-symmetricalthough the geometry of the combustion chamber is not. It means that the shape of the forced flameis mostly imposed by the injector which is axi-symmetric. The 2D plane cut colored by the responseamplitude is separated in two zones which react differently to the excitation.• In Zone 1, the response amplitude is intense. It corresponds to high values of local heat release132

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)fluctuations. This is due to the equivalence ratio of the axial swirler (φ = 0.8) which is higherthan the equivalence ratio of the diagonal swirler (φ = 0.5). The mean heat release in this zone istherefore higher and leads to more heat release fluctuations for a given incoming velocity fluctuation.• In Zone 2 the level of mean heat release is almost constant along the flame front but the sideanchored at the burner mouth and the side stabilized by the precessing vortex core [47, 147] showdifferences in their response amplitude to the fluctuation. The part of the flame front attached tothe outer part of the burner reacts more weakly to the perturbation showing that this part of theflame is well anchored and less prone to instability.The separation of the response of the white noise forced flame in two zones is not so clear on figure 4.34: some parts of the zone 2 show a response which has the same amplitude as in zone 1. The responseamplitude is not perfectly axi-symetric. This can be due to the effects of the geometry in the case of thewhite-noise forced flame. Also note that as for the harmonically forced case, the outer flame front reactsless intensively to the incoming perturbation than the inner side.133

TRANSFER FUNCTIONS OF FLAMESFigure 4.33 - 2D longitudinal cut colored with the amplitude of FTF: HF-FFT method. Black line marks themean flame position.Figure 4.34 - 2D longitudinal cut colored with the amplitude of FTF: WN-WH method. Black line marks themean flame position.134

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)Time delayFigures 4.35 and 4.36 show 2D longitudinal plane cuts colored by the time delay of the FTF at 90Hzrespectively obtained using the HF-FFT and WN-WH methods. The delays change between 0 and 11 ms(the forcing period at 90Hz).The zones described for the amplitude analysis are also present on those figures. They also correspondto different types of responses:• In zone 2, the behaviour of the time delay for the present flame is well described by the 1D analysis,with lower time delays after the mean flame front and higher time delays in the inner fluctuatingarea.• In zone 1, the distribution of time delays is reversed: shorter time delays are observed before themean flame front. It is due to the fact that the flame first contracts in the cycle, burning nearer tothe injection, and then expands to reach the black zone which corresponds to higher time delays.The dimensions of the two zones are different whether the flame is forced with a harmonic signal or withfiltered white noise. It shows that even if the global time delay is almost the same for the two forcingmethods, the influence of the other frequencies present with WN-WH method changes the local responseof the flame (i.e. the localisation of zones 1 and 2) at 90Hz.The maps of amplitudes (Fig.(4.33) and (4.34)) and of local time delays (Fig.(4.35) and (4.36)) exhibitobvious structures. Their interpretation however is difficult. Zone 1 seems to be the place where the mostintense interaction takes place, while zone 2 yields more diffuse flame responses but with time delayswhich exhibit the pattern of oscillating flames seen in Fig.(4.5) for 1D flames. This suggests that theinner flame, stabilized on the axial swirler hub, is the most unstable one and the first source of the flameresponse. Being more precise is difficult and using these results to study stability requires an acousticcode and is done in other PhDs [8].Weighted Probability density functions of time delays.The 3D fields of amplitude and time delay obtained with both methods are used to reconstruct theWeighted-PDF of time delays. The way these PDF of time delays are obtained is described in section4.3. Figure 4.37 compares the two weighted PDF of time delays. Both show a maximum around2 ms which is the global value of the time delay at 90Hz. They also both show a peak around 6ms. Itmeans that even if the global value of the time delay is 2ms a non negligible part of the fluctuations ofheat release that occur out of phase with the inlet velocity. Figure 4.35 shows that this value correspondsto heat release fluctuations occurring inside the mean flame position in zone 1 and 2. The main differencebetween the two pdf of time delays is the probability of finding a time delay around 1 ms which is muchsmaller when the flame is forced with filtered white-noise. This may be due to the influence of higherfrequencies on the 90Hz turbulent FTF using WN-WH method.135

TRANSFER FUNCTIONS OF FLAMESFigure 4.35 - 2D longitudinal cut colored with the time delay of FTF: HF-FFT method. Black line marks themean flame position.Figure 4.36 - 2D longitudinal cut colored with the time delay of FTF: WN-WH method. Black line marks themean flame position.136

4.4 Configuration D : turbulent burner in a 15 ◦ sector (3D)Figure 4.37 - PDF of weighted time delays : WN-WH method. HF-FFT method.4.4.5 Concluding remarks for configuration DThe influence of frequency on the FTF of a turbulent combustor has been studied using three differentmethods of forcing and post-processing. The first output of this study concerns the global FTF comparison.The WN-WH method gives results that are similar to the results obtained using the harmonicmethod though its computational cost is much lower (4 times in this case). Yet, it has to be noticed thatthe white-noise forcing method requires the use of the Wiener-Hopf relation inversion and therefore hasspecific drawbacks that have to be carefully addressed.• First the WH method is very sensitive to the size of the filter used for the inversion. The numberof elements of the filter multiplied by the time sampling of the signal represents the time memoryof the system. When inverted, this time represents the minimum frequency this method will beable to treat. But due to the matrix inversion, the size of the filter also has to remain as low aspossible to give precise results. This leads to an optimal value of the size of the filter which maybe sometimes hard to find.• When forcing with white noise, extreme care has to be given to the filter cut-off frequency and tothe type of filter used. In this study, the cut-off frequency is set to 600Hz which has been shown astoo close to the frequencies of interest. A fourth-order low-pass filter has also been used to avoidany unwanted high frequency influences on the measured FTF.The local response of the turbulent flame is studied at 90Hz. Both methods (HF-FFT and WN-WH) givesimilar FTF and probability density functions of time delays. They both show that as already observed inchapter 4.3 the local response of the turbulent flame can be very different from its global response, withlocations where the fluctuations of HR can be out of phase with the fluctuations of velocity even thoughthe global response is almost in phase.137

TRANSFER FUNCTIONS OF FLAMES4.5 Evaluation of FTF measurements methods in LES.Four different methods for the measurement of FTF in LES have been tested. Table 4.6 summarizes themain advantages and drawbacks of these different methods that derive from the previous results. + and- signs under the name of each method is the global author evaluation for their use in the measument ofFTF in LES. Signs ” ! ” emphasize the major advantages and drawbacks of the diffrent methods.An important aspect of this study is their link with stability analysis of combustors. Obviously,FTF do have an influence on the frequency and amplification rates of modes in the numerical methodsused for combustor stability [9, 91]. The present results show that methods like WN-WH and HF-FFTprovide slightly different local FTF maps. Whether these differences will affect or not significantly thefrequencies and amplification rates of modes remains to be studied. What this study has shown is howto construct FTFs which is the important ”brick” of acoustic analysis : further studies are needed todetermine which method will be the most precise. Note also that experimental results on global FTF areavailable but that no-one has studied local FTF yet. This is obviously a required step for the future.138

4.5 Evaluation of FTF measurements methods in LES.Methods Advantages DrawbacksTD method- -• Computational cost: Only requires the simulationof the steady state solution.• Complexity: Injection of a passive scalar to get theflow time history• ! Results: Only recovers time delays• ! Precision: Lack of precision when time delay isnot directly linked with convective phenomena• ! Precision: compares well with experimentsin terms of time delaysHF-FFT method+ +• ! Precision: compares well with analyticalmodels• ! Computational cost: each frequency requires adifferent computation• Results: gives access to the local response ofthe flameWN-FFT method- -• Computational cost: only one computationof the white-noise forced configuration isneeded• Precision: lack of precision due to FFT postprocessingof non-harmonic signal on configurationB• ! Results: Totally incoherent results for configurationDWN-WH method+ + (+)• ! Computational cost: only one computationof the white-noise forced configuration isneeded• ! Precision: compares well with HF-FFT results• Results: gives access to the local response ofthe flame• Precision: results are dependent to the L parameterfor the Wiener-Hopf inversion• ! Complexity: difficulties in finding the optimalvalue of the L parameter.Table 4.6 - Comparison of the advantages and drawbacks of TD,HF-FFT,WN-FFT and WN-WH methods.139

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