M Gatti

R Gaudron

C Mirat

L Zimmer

T Schuller

M Merk

C Silva

W Polifke

Impact of swirl and blu-body on the transfer function of premixed ames

deserve a great thank.

I also want to thank all the researchers and Ph.D. students of the EM2C laboratory and at TU Darmstadt, for creating such good workplace environments.

At last, the biggest thanks of all goes to my family, for having supported me during all these years. In particular I want to thank Gloria for

I want to thank all the members of the jury for accepting to review my thesis manuscript. A big thank goes to my thesis directors, starting from Thierry Schuller, that four years ago let me start my Ph.D. at the EM2C Laboratory, despite my lack of knowledge of any laboratory equipment at all. He has always been supportive and comprehensive, even in those moments when it would have been easier to be angry or disappointed. Most of all, I appreciated his enthusiastic approach to bring out the best in each of his students. I thank Clément Mirat for the great help that he gave me to solve all the technical problems I encountered. He taught me all the aspect of conducting laboratory experiments and he was always available when I needed help. I also thank Laurent Zimmer for teaching me laser diagnostics, from application to post-processing. I really valued the chance to work with such an expert in this eld. I want to thank Prof. Amsini Sadiki for welcoming me at TU Darmstadt during my secondment period in Germany and for the fruitful discussions we had about my manuscript.

Renaud Gaudron was the person I spent most of my time with during my three years at the EM2C laboratory, considering that we shared both the same experimental room and the same oce. I have to and I am very happy to thank him, not only for his guidance during my rst weeks at the lab, for showing me a lot of interesting engineering tricks, for teaching me a bit of the french language, for a lot of interesting discussion related and not related to the job, for the help he gave me in preparing articles and presentation for conferences, but mostly for coping with me despite my terrible taste in music.

None of the work done in this thesis would have been possible without the help from all the technical and administrative sta of the EM2C laboratory, that

The analysis proceeds by examining mechanisms leading to FTF maximum and minimum gain values, by conducting a detailed experimental and a numerical analysis of the ow and ame dynamics at these specic forcing frequencies. It is shown that the maximum response of swirled ames, stabilized by a central blu-body, is controlled by the formation of large vortical structures in the external shear layer of the ow that later rollup the ame tip. At the minimum ame response, this mechanism of ame rollup around a vortex is strongly attenuated, mainly due to the absence of large vortical structures shed from the burner lip. It is demonstrated experimentally and conrmed by numerical simulations, that this is not the result of interfering mechanisms between the vortex growth and the apping motion of the ame, since these vortical structures are already absent in the cold ow.

The cold ow dynamics is shown to be controlled by complex interactions between azimuthal vortical waves and acoustic longitudinal waves. These waves combine, leading to large swirl number uctuations by well described mechanisms, but they also lead to a strong reduction of toroidal vortical structures of the cold ow response by a mechanism that remains to be fully elucidated. This, in turn, leads to the conclusion that the strong reduction of the FTF gain of swirled ames, stabilized by a central blu body, is only the consequence of the cold ow injector dynamics. This opens the path to new ways of controlling the response of these ames.

As a conclusion, three main mechanisms determining the response of LP swirled ames to acoustic waves were identied in this work. They are all associated to the cold ow dynamics of the injection unit. The rst one, leading to high FTF gains, is the formation of large toroidal vortical structures in the external shear layer of the swirled ow. The second one, possibly leading to reductions of the FTF gain, is the interference of longitudinal acoustic waves and azimuthal convective waves generated by the swirling vane and leading to large swirl number oscillations. The last one is the response of the internal recirculation bubble to acoustic forcing, that exhibits large axial displacements for aerodynamically stabilized ames. Controlling the dynamics of these three mechanisms, by analyzing the cold ow injector response to sound waves, appears as a promising way to shape the FTF of LP swirled ames.

Résumé

Les systèmes de combustion à prémélange pauvre (PP), qui brûlent du gaz naturel avec un excès important d'air par rapport à la stoechiométrie, sont l'une des technologies les mieux adaptées pour la réduction des émissions de gaz à eet de serre et des émissions de polluants résultant de la combustion de ressources fossiles. Malheureusement, ces technologies sont très sensibles aux phénomènes dynamiques. L'amélioration des performances des systèmes PP est entravée depuis de nombreuses années par leur vulnérabilité aux phénomènes d'extinction, aux retours de amme (ashback) dans l'injecteur et aux instabilités de combustion. La plupart des chambres de combustion des turbines à gaz utilisent des vrilles pour mettre en rotation l'écoulement. On parle de tourbillonneurs ou de swirleurs pour stabiliser des ammes compactes et permettre une combustion ecace et propre avec des densités de puissance élevée. Une meilleure connaissance des mécanismes de la dynamique de la combustion d'écoulements swirlés PP présente un intérêt aussi bien pratique que fondamental.

Le brûleur Noisedyn a été spécialement conçu lors de ces travaux pour répondre à cet objectif. La géométrie de ce brûleur peut être facilement modiée pour tester les eets d'une série de modications du système d'injection sur l'état de la combustion. Parmi ceux-ci, les eets d'un swirleur annulaire et d'un insert sur l'axe de l'injecteur sont examinés. L'analyse débute par un examen des eets du swirleur radial équipant le système d'injection avec un insert central.

Le nombre de swirl S est connu pour contrôler la topologie de l'écoulement résultant et, à cet égard, il contrôle également la stabilisation de la amme. Pour un faible niveau de swirl (S < 0.50 dans ce travail), les ammes prennent une forme allongée et sont proches de l'extinction, alors que pour un niveau de swirl excessif (S > 0.80), le risque de ashback augmente considérablement. Pour des nombres de swirl intermédiaires, des ammes compactes bien stabilisées sont obtenues dans le foyer Noisedyn.

Le nombre de swirl S détermine également largement la réponse des ammes swirlées aux perturbations acoustiques. Lorsque le niveau de swirl est faible, la fonction de transfert de amme (FTF) se rapproche de celle d'une ammes x Résumé stabilisée dans le sillage de l'insert central avec des valeurs de gain élevées dépassant l'unité sur une large plage de basses fréquences. Au fur et à mesure que le niveau de swirl augmente, les FTF sont caractérisées par un motif en dents de scie, avec une réponse maximale directement suivie d'une chute importante de cette réponse dans la même gamme de fréquences. Il est également montré qu'une réduction régulière de la réponse minimale de la amme est obtenue lorsque le nombre de swirl augmente progressivement, jusqu'à un niveau supérieur au seuil au-dessus duquel un retour de amme dans l'injecteur déclenché par le forçage acoustique est observé pour les ammes fortement swirlées. Les modications de la vitesse d'injection du mélange combustible ou de la position de l'aube tourbillonnante, tout en maintenant un nombre de swirl constant, produisent un décalage en fréquence des extrema de gain de la FTF sans modier les valeurs de ces extrema. Ces informations sont utilisées pour redimensionner la FTF des ammes swirlées en fonction d'un nombre de Strouhal basé sur la vitesse d'injection et sur une longueur eective incluant la longueur de la amme et la distance de la vrille à la sortie de l'injecteur.

Il est montré qu'un bon réglage de l'unité d'injection, ainsi que des modications des propriétés acoustiques de la chambre de combustion, par exemple en changeant la longueur du tube d'échappement, permettent de réduire ou de déclencher des instabilités thermo-acoustiques observées dans la chambre de combustion NoiseDyn. En particulier, lorsqu'une forte instabilité est observée à une fréquence correspondant à une réponse élevée de la FTF, les oscillations peuvent être amorties en ajustant le système d'injection de sorte à obtenir une réponse faible de la FTF à cette même fréquence.

L'analyse procède ensuite à l'examen des mécanismes conduisant aux valeurs de gain maximum et minimum de la FTF grâce à une étude expérimentale et numérique fouillée de la dynamique de l'écoulement et de la amme à ces fréquences de forçage spéciques. On montre que la réponse maximale des ammes swirlées stabilisées par un insert central est contrôlée par la formation de grandes structures tourbillonnaires dans la couche de cisaillement externe du jet issu de l'injecteur. Ces structures sont convectées et enroulent ensuite la pointe de la amme. Au minimum de réponse de la FTF, ce mécanisme d'enroulement de la amme autour d'un vortex est fortement atténué, principalement en raison de l'absence de grandes structures tourbillonnaires libérées au niveau des lèvres du brûleur. Il est démontré expérimentalement et conrmé par des simulations numériques que cela n'est pas le résultat de mécanismes d'interférence entre la croissance du vortex et le mouvement de battement de la amme, car ces structures tourbillonnaires sont également absentes dans l'écoulement sans réaction de combustion.

La dynamique de l'écoulement à froid est contrôlée par des interactions complexes entre des ondes de vorticité azimutales et des ondes acoustiques longi-Résumé xi tudinales. La combinaison de ces ondes conduit à des uctuations importantes du nombre de swirl par des mécanismes désormais bien décrits, mais elles entraînent également une forte réduction des structures tourbillonnaires toroïdales libérées dans l'écoulement par un mécanisme qui reste à élucider complètement. Ceci conduit à conclure que la forte réduction du gain de la FTF des ammes swirlées stabilisées par un corps central résulte uniquement de la dynamique de l'injecteur à froid. Modier cette dynamique en agissant sur l'injecteur ouvre la voie à de nouvelles façons de contrôler la réponse de ces ammes.

La dernière contribution de ce travail consiste à examiner le rôle de l'insert central dans ce processus. Des expériences sont menées sur des ammes présentant le même niveau de swirl, la même richesse et la même vitesse d'injection du mélange combustible, mais l'une est ancrée sur le corps central de l'insert tandis que l'autre est stabilisée aérodynamiquement en retirant ce composant et en réduisant légèrement le diamètre du tube d'injection. On constate que la forte réduction du gain de la FTF observée pour la amme stabilisée swirlée avec un insert central n'est pas présente dans les résultats pour la FTF de la amme entièrement stabilisée aérodynamiquement. Il est montré dans ce dernier cas que la dynamique de la région de recirculation interne présente de grandes oscillations axiales et dière radicalement de celle de la amme stabilisée par le swirl et l'insert central. La formation de structures tourbillonnaires dans la couche de cisaillement externe du jet froid issu de l'injecteur est également fortement modiée en raison de l'oscillation de la zone de recirculation centrale. Ces éléments indiquent que la réponse à froid d'injecteurs swirlées avec et sans corps central dière également considérablement.

En conclusion, trois mécanismes importants ont été identiés dans ce travail qui déterminent la réponse de ammes swirlées aux ondes acoustiques. Ils sont tous associés à la dynamique de l'écoulement à froid à travers le système d'injection. Le premier qui se traduit par des gains élevés de FTF est la formation de grandes structures tourbillonnaires toroïdales dans la couche de cisaillement externe du jet swirlé. Le second qui est associé à un processus de réduction de gain de la FTF sont des mécanismes d'interférence entre les ondes acoustiques longitudinales et des ondes de vorticité azimutales générées par le tourbillonneur, ce qui entraîne de fortes oscillations du nombre de swirl. Le dernier est la réponse de la bulle de recirculation centrale au forçage acoustique. Celle-ci présente de grands déplacements axiaux pour les ammes stabilisées aérodynamiquement. Contrôler la dynamique de ces trois mécanismes en analysant la réponse des injecteurs aux ondes sonores de l'écoulement à froid apparaît comme un moyen prometteur pour modeler la forme des FTF de ammes swirlées PP.

Zusammenfassung

Magervorgemischte (im Englischen, Lean Premixed (LP)) Brennkammern, in denen Erdgas und Luft in einem Äquivalenzverhältnis weit unter dem stöchiometrischen Wert verbrannt werden, sind eine der bekanntesten Technologien zur Reduzierung der Treibhausgasemissionen und der Schadstoemissionen bei der Verbrennung fossiler Brennstoe. Leider sind diese Technologien sehr empndlich gegenüber dynamischen Phänomenen. Leistungs-verbesserung solcher LP Systeme wird seit vielen Jahren durch ihre Anfälligkeit für Blowout, Flashback und Verbrennungsinstabilitäten erschwert. Die meisten LP-Gasturbinenbrenner verwenden verdrallte Strömungen, um kompakte Flammen ezient und better bei hohen Leistungsdichten zu stabilisieren. Bessere Kenntnis der Mechanismen der instationären Verbrennungsvorgänge in magervorgemischt verdrallten Flammen ist daher von groÿer praktischer Bedeutung.

In diesem Zusammenhang wurde der Noisedyn-Brenner im Rahmen dieser Arbeit speziell entwickelt. Dabei wurde der Brenner so konzipiert, dass leichte Änderungen der Injektor-Geometrie ermöglicht werden, um ihre Auswirkungen auf die Verbrennungseigenschaften zu evaluieren. Es wurden die Wirkung einer ringförmigen verdrallten Schaufel (Vane) und der Einuss eines zentralen Staukörpers untersucht.

Zunächst wurde die Untersuchung der Eekte der radial verdrallten Schaufel der mit einem zentralen Staukörper ausgestatteten Injektionseinheit durchgeführt. Da die Drallzahl S die Topologie der verdrallten Strömung und in dieser Hinsicht auch die Flammenstabilisierung determiniert, wurde die Drallzahl variiert. Bei nidriegen Drallzahlen (S<0.50) brennen die Flammen nahe des Blowos und weisen eine löngliche Form auf. Bei sehr hohen Drallzahlen (S>0.80) nimmt die Gefahr eines Rückschlags/Flashbacks drastisch zu, während gut stabilisierte kompakte Flammen bei mäÿigen Drallzahlen erhalten werden.

Die Drallzahl S bestimmt auch maÿgeblich die Antwort der verdrallten Flammen auf akustische Störungen. Es konnte gezeigt werden, dass bei niedrigen Drallzahlwerten die Flammenübertragungsfunktion (im Englischen "Flame-Transfer-Function" (FTF)) nähert sich diejenige von Keilammen mit groÿen Verstärkungswerten über Eins bei niedrigen Frequenzen. Mit zunehmender Drallstärke zeichnen sich die FTF durch ein Sägezahnmuster aus, welche durch eine maximale Antwort charakterisiert ist, die unmittelbar durch einen starken Druckabfall im gleichen Frequenzbereich gefolgt wird. Es wurde auch festgestellt, dass eine regelmäÿige Reduzierung der Flammenantwort erzielt wird, wenn die Drallzahl schrittweise bis zu einem Schwellenwert erhöht wird. Über diesen Wert wird ein Flashback ausgelöst, welcher durch eine akustische Anregung bei stark verdrallten Flammen zu beobachten ist. Die hier gewonnene Information wurde verwendet, um die FTF von verdrallten Flammen in Abhängigkeit von der Strouhal-Zahl auf der Grundlage der Strömungsgeschwindigkeit und einer eektiven Länge einschlieÿlich der Flammenlänge und des Abstands zwischen dem Swirler und dem Injektorauslass neu zu skalieren.

Es wurde gezeigt, dass eine richtige Einstellung der Änderung der Injektoreinheit zusammen mit Modikationen der akustischen Eigenschaften der Brennkammer, z. B. durch eine Änderung der Länge des Abgasrohrs, können thermoakustische Instabilitäten, die in der NoiseDyn-Brennkammer beobachtet wurden, passiv kontrolliert werden. Insbesondere wenn eine starke Instabilität bei einer bestimmten Frequenz beobachtet wird, die einer hohen FTF-Flammenantwort entspricht, können Oszillationen gedämpft werden, indem das Injektorsystem so eingestellt wird, dass es bei dieser spezischen Frequenz eine niedrige Flammenantwort aufweist.

Diese Analyse wurde fortgesetzt, indem Mechanismen untersucht wurden, die zu maximalen und minimalen FTF-Werten führen. Hierzu wurden detaillierte experimentelle und numerische Analyse der Strömungs-und Flammendynamik bei diesen spezisch erzwingenden Frequenzen durchgeführt.

Es wurde gezeigt, dass eine maximale Antwort der verdrallten Flammen, die durch einen zentralen Staukörper stabilisiert werden, durch die Bildung groÿer Wirbelstrukturen in der äüÿeren Scherschicht der Strömung gesteuert wird, welche sich später in die Flammenspitze entwicklen. Bei einer minimalen Flammenantwort wird dieser Mechanismus stark gedämpft, hauptsächlich aufgrund der nicht auftretenden groÿen Wirbelstrukturen. Experimentell wurde gezeigt, und durch numerische Simulationen bestätigt, dass das Ergebnis nicht von Interferenzmechanismen zwischen dem Wirbelwachstum und der Flappingbewegung der Flamme zurückzuführen ist, da die Wirbelstrukturen bereits in der kalten Strömung nicht auftreten.

Es wurde ferner gezeigt, dass die Dynamik der kalten Strömung durch komplexe Wechselwirkungen zwischen azimuthalen Wirbelwellen und akustischen Longitudinalwellen gesteuert wird. Zusammen führen diese Wellen durch gut beschriebene Mechanismen, zum einen, zu groÿen Schwankungen der Drallzahlen, und zum anderen, zu einer starken Reduzierung der toroidalen Wirbel-xv strukturen der Antwort der kalten Strömung durch einen Mechanismus, der noch vollständig aufgeklärt werden muss. Es lieÿ sicht schlussfolgern, dass die starke Reduzierung der FTF-Verstärkung von verdrallten Flammen, die durch einen zentralen Staukörper stabilisiert wird, nur die Folge der Dynamik des Injektors der kalten Strömung ist. Dies önet den Weg zu neuen Möglichkeiten, die Antwort der Flammen zu kontrollieren.

Der letzte Beitrag dieser Arbeit bestand in der Untersuchung der Rolle des zentralen Staukörpers in diesem Prozess. Es wurden Versuche mit zwei Flammen durchgeführt, die den gleichen Drallgrad, das gleiche äquivalenzverhältnis und die gleiche Strömungsgeschwindigkeit aufweisen. Jedoch wurde eine Flamme an einem zentralen Staukörper verankert, während die andere Flamme durch Entfernen dieser Komponente und leichtes Verringern des Durchmessers des Einspritzrohrs vollständig aerodynamisch stabilisiert wurde. Es konnte festgestellt werden, dass die starke Verringerung der FTF-Verstärkung, welche für die durch Staukörper verdrallt stabilisierte Flamme beobachtet wurde, bei der vollständig aerodynamisch stabilisierten Flamme nicht stattndet.

In diesem letzteren Fall wurde gezeigt, dass die Dynamik des internen Rezirkulationsbereichs, der groÿe axiale Schwankung aufweist, unterscheidet sich drastisch von derjenigen der mit Staukörpern stabilisierten Drallamme. Die Bildung von Wirbelstrukturen in der äuÿeren Scherschicht verändert sich stark aufgrund der Oszillation der zentralen Wirbelblase, was darauf hinweist, dass sich die Antwort der kalten Strömung aus den verwirbelten Injektoren mit und ohne Zentralkörper ebenfalls drastisch unterscheidet.

Zusammenfassend wurden in dieser Arbeit drei Hauptmechanismen identiziert, die die Antworte von LP-Flammen auf akustische Wellen bestimmen. Sie hängen alle mit der Dynamik der kalten Strömung der Injektoreinheit zusammen. Die erste, die zu hohen FTF-Verstärkung führt, ist die Bildung groÿer toroidaler Wirbelstrukturen in der äuÿeren Scherschicht der verdrallten Strömung. Die zweite, die möglicherweise zu einer Verringerung der FTF führt, ist die Interferenz von longitudinalen akustischen Wellen und azimuthalen Konvektionswellen, die von der Wirbelschaufel erzeugt werden und zu Schwankungen mit groÿer Drallzahl führen. Die letzte ist die Antwort der internen Rezirkulationsblase auf akustische Störungen, die groÿe axiale Verschiebungen für aerodynamisch stabilisierte Flammen aufweisen. Das Kontrollieren der Dynamik dieser drei Mechanismen durch eine Analyse der Antwort des kalten Strömungsinjektors auf akustischen Wellen scheint eine vielversprechende Möglichkeit zu sein, die FTF von LP verdrallten Flammen zu modellieren bzw. zu formulieren.

Introduction Context

Between 1971 and 2015, the world energy production and consumption more than doubled (IEA, 2017a). The development of renewable energy sources remains too slow and the combustion of fossil fuels still accounts for more than 80% of this production in 2015 (Fig. 1) and this trend is not expected to abruptly change in the next decades (IEA, 2017b). A well known drawback of the combustion of hydrocarbon fuels is the production of greenhouse gases, since carbon dioxide CO 2 is the main product of the combustion reaction, together with water vapor. Combustion also produces traces of other pollutants, like unburned hydrocarbons HC, nitric oxides NO x and carbon monoxide CO. The requirements to reduce pollutant emissions are more and more stringent and actions against the climate change trend, caused by fossil fuel burning, are urgently needed. On 12 December 2015, as an outcome of the 2015 United Nations Climate Change Conference, the 196 countries involved signed the Paris Agreement. For the rst time, all the nations involved were brought into a common cause to undertake ambitious eorts to combat climate change and adapt to its eects. The central aim is to strengthen the global response to Typical CO and NOx production from lean premixed combustion of methane. Adapted from [START_REF] Brown | Combustion control apparatus and method[END_REF] the threat of climate change, by keeping a global temperature rise this century well below 2 degrees Celsius above pre-industrial levels and to pursue eorts to limit the temperature increase even further to 1.5 degrees Celsius. In this sense, the advantages of natural gas compared to other fossil fuels are well known. For example CO 2 emissions (per unit of Joule produced) from natural gas are on average around 40% lower than coal combustion and around 20% lower than burning of oil (IEA, 2016). Due to this environmental benets and also to its widespread availability, the share of natural gas among other fossil fuels has increased in these last decades, as illustrated in Fig. 1.

To cope with the increasingly strict regulations for pollutant emissions, with particular attention to NO x production [START_REF] Correa | A review of NOx formation under gas-turbine combustion conditions[END_REF][START_REF] Bahr | Aircraft Turbine Engine NOx Emission Abatement[END_REF], the gas turbine industry moved towards new combustion technologies, e.g. staged combustion, oxy-fuel combustion or ameless combustion. Among these technologies, lean premixed (LP) combustion oers several advantages. In LP systems, fuel and air are premixed before entering the burner at a well controlled equivalence ratio, to avoid the formation of stoichiometric regions. The combustion zone is operated with a large excess of air, with respect to stoichiometric values, to reduce the ame temperature. The typical evolution of main pollutant formations for a lean mixture of methane, which is the main compound of natural gas, and air is shown in Fig. 2. NO x and CO emissions follow opposite trends. On one hand, at low ame temperatures or equivalence ratios, NO x production is low, but it then increases exponentially with temperature. On the other hand low temperatures prevent the complete oxidation of CO and generate unburned HC. In an ideal case, one tries to drive the system towards the optimum point where both CO and NO x levels will be minimum. In an industrial context, the objective is to cope with emission limits imposed by the legislation, but one is limited by the machine performance that needs to deliver the desired power, preserve the components integrity and be as versatile as possible. Therefore, the control system has to operate the combustor in a range of equivalence ratios avoiding the rapid increase of CO and NO x [START_REF] Docquier | Combustion control and sensors: a review[END_REF] among many other factors altering its behavior.

Lean premixed swirled injectors

The stabilization of ames in LP systems and many other modern combustion systems is achieved by imparting an azimuthal component to the ow, usually by passing the air or the combustible mixture stream through a swirling vane or a set of swirlers [START_REF] Candel | Progress and challenges in swirling ame dynamics[END_REF]. The swirl intensity is usually characterized by the swirl number S, dened as the ratio of the axial ux of the tangential momentum to the product of the axial momentum ux and a characteristic radius R of the injector [START_REF] Gupta | Swirl ows[END_REF]. By neglecting the pressure term in the axial momentum ux [START_REF] Beer | Combustion Aerodynamics[END_REF], the swirl number is dened as:

S = 1 R u θ u z r 2 dr u 2 z rdr . (1)
where u z and u θ are the axial and azimuthal velocity components, r the radius and R a characteristic outer radius. When a sucient amount of rotation is imparted to the ow, a high adverse axial pressure gradient is achieved which leads to a ow reversal and generates a Central or Inner Recirculation Zone (IRZ) [START_REF] Huang | Dynamics and stability of lean-premixed swirl-stabilized combustion[END_REF]. Figure 3 shows a comparison of the streamline patterns and mean temperature elds of a low (S < 0.6) and an high (S > 0.6) swirling ame. In both cases, an Outer Recirculation Zone (ORZ) is present due to the sudden expansion of the cross section area when the ow leaves the burner and enters the combustion chamber. In the low-swirl number case with S = 0.44 in Fig. 3, the recirculating ow in the central region is mainly due to a small wake recirculation zone (WRZ) behind the blu-body. In the absence of swirl, only these two zones would exist. With the addition of swirl, vortex breakdown takes place (Fig. 4) for a suciently large swirl level and leads to the formation of the IRZ.

As the swirl number exceeds a threshold, generally taken equal to S > 0.60 [START_REF] Lucca-Negro | Vortex breakdown: A review[END_REF], the inner recirculation zone IRZ moves upstream and merges with the wake recirculation zone WRZ. The ow region of vortex breakdown provides the dominant ame stabilization mechanism. Thanks to this mechanism, LP systems can sustain compact ames in high speed ows even under lean conditions [START_REF] Huang | Eect of swirl on combustion dynamics in a lean-premixed swirl-stabilized combustor[END_REF].

Dynamical phenomena, however, hinder the performance of combustors equipped with swirled injectors. At rst, these systems are often operated near the lean blowo limit, meaning that a small uctuation of the owrate could cause ame extinction. Secondly, when operated at larger equivalence ratios, a small ow oscillation could cause the IRZ to enter the injector, possibly resulting in ame ashback. Among these problems, that of combustion instabilities has drawn a considerable amount of attention since last century [START_REF] Crocco | Aspect of Combustion Stability in Liquid Propellant Rocket Motors Part I[END_REF] and still constitutes a dicult challenge for designers [START_REF] Keller | Thermoacoustic oscillations in combustion chambers of gas turbines[END_REF][START_REF] Lieuwen | Combustion Instabilities: Basic Con-References[END_REF][START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF]. LP systems are particularly prone to this problem. One of the major dierences in design, between lean-premixed and diusioname type combustion systems, is the absence in the former of perforated liners around the combustor [START_REF] Candel | Progress and challenges in swirling ame dynamics[END_REF]. In diusion controlled combustion burners, a substantial fraction of air enters through the liner. This not only help in decreasing the temperature of the combustion products before entering the turbine, but also serves as damping of acoustic waves [START_REF] Keller | Thermoacoustic oscillations in combustion chambers of gas turbines[END_REF]. Acoustic damping is reduced in LP systems because the perforated liners are mostly eliminated in modern premixed combustors, increasing their sensitivity to acoustic feedback and combustion instabilities.

Combustion instabilities

Extensive review of combustion instability phenomena can be found in the literature, from fundamental mechanisms [START_REF] Candel | COMBUSTION DYNAMICS AND CONTROL : PROGRESS AND CHALLENGES[END_REF] to application for real engines [START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF]. Only a very brief presentation is provided in this section, while readers are referred to these references and to the textbooks from Lieuwen andYang, 2005 and[START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF][START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF] for more details. Thermoacoustic instabilities results from the complex dynamic interactions between acoustic waves and unsteady heat release (Fig. 5). When the heat release is provided by combustion, they are also designated as combustion instabilities.

They can cause structural vibrations, augmented heat uxes to the chamber walls, ame quenching or ashback and often lead to severe damage [START_REF] Lieuwen | Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling[END_REF]. Combustion instabilities can appear in almost any combustion system and have been observed in the development of propulsion systems, like liquid rocket engines, solid rocket motors, ramjet and scramjet engines and many domestic and industrial boilers or furnaces [START_REF] Huang | Dynamics and stability of lean-premixed swirl-stabilized combustion[END_REF].

Flow/mixture perturbations

Heat release oscillations

Acoustic oscillations

Figure 5: Feedback loop between acoustics and combustion leading to combustion instabilities. Adapted from [START_REF] Lieuwen | Combustion Instabilities: Basic Con-References[END_REF] This phenomenon has been known for more than a century. Rayleigh, 1878, rst stated the conditions under which unsteady heat release adds energy to the acoustic eld: if heat is released when a pressure oscillation is near its maximum, the pressure uctuation increases in amplitude; on the other hand, if heat release occurs when the pressure oscillation is at a minimum point, the pressure uctuation is attenuated. This is a necessary but not sucient condition for combustion instabilities to arise, since the energy added to the acoustic eld by combustion must overcome that dissipated within the burner and/or transmitted through its boundaries [START_REF] Cantrell | Interaction between Sound and Flow in Acoustic Cavities: Mass, Momentum, and Energy Considerations[END_REF]. Mathematically, this criterion can be expressed as an acoustic energy balance [START_REF] Dowling | FEEDBACK CONTROL OF COMBUSTION OSCILLATIONS[END_REF][START_REF] Nicoud | Thermoacoustic instabilities: Should the Rayleigh criterion be extended to include entropy changes?[END_REF][START_REF] Durox | Experimental analysis of nonlinear ame transfer functions for dierent ame geometries[END_REF][START_REF] Brear | Disturbance energy transport and sound production in gaseous combustion[END_REF]:

γ -1 ρc 2 1 T V T p q dtdV ≥ 1 T ∂V T p u dtdS + 1 T V T ζ a dtdV (2)
where p , u , q , V , ∂V , T and ζ a are, respectively, the acoustic pressure, velocity and volumetric rate of heat-release oscillations, volume and volume boundaries, period of oscillations, and acoustic energy loss processes within the ow (see [START_REF] Howe | Acoustics of Fluid-Structure Interactions[END_REF].

Pressure uctuations indirectly lead to perturbations of the heat-release through three main mechanisms [START_REF] Ducruix | Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms[END_REF]:

• Equivalence-ratio oscillations: this can happen when the air and fuel injection line respond dierently to the pressure perturbation. A mixture of oscillating equivalence ratio is then convected to the ame producing the heat-release oscillation. This mechanism does not concern fully premixed systems featuring a common air-fuel injection line. • Flame surface variation: pressure perturbations are associated with acoustic velocity oscillations that modify the ame surface and thus the heat release. • Flame-vortex interaction: when the acoustic velocity oscillation reach the burner rim, vortical structures are generated and then convected towards the ame. The interaction rapidly changes the ame surface area and thus the heat release.

Some methods were developed during the years to suppress or attenuate combustion instabilities and can be divided into active or passive [START_REF] Mcmanus | A review of active control of combustion instabilities[END_REF][START_REF] Candel | COMBUSTION DYNAMICS AND CONTROL : PROGRESS AND CHALLENGES[END_REF][START_REF] Lieuwen | Combustion Instabilities In Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling[END_REF].

In active control techniques, the instability is dynamically detected and corrected using a feedback control loop to modify one or more input parameters.

Passive control involves changes of fuel or geometry designs (for example, in the composition or types of reactants, fuel injection devices and chamber geometry, or the installation of acoustic dampers), either to reduce the rate at which energy is transferred to unsteady motions (reduce the left term in Eq. ( 2)), or to increase losses of energy (increase the right term in Eq. ( 2)), such as by the use of suitable resonators to introduce a dissipative process. One of the earliest examples of passive control was applied during the development of the F1 rocket engine [START_REF] Oefelein | Comprehensive Review of Liquid-Propellant Combustion Instabilities in F-1 Engines[END_REF]. The strong instabilities observed during the design process were eventually suppressed by substantial modications of the injection system geometry. But this was obtained after a costly trial and error process with almost 2000 full-scale tests.

Much progress has been made in the eld of combustion instabilities since that experience. It is nowadays possible to predict the thermoacoustic stability of practical systems, at a limited computational cost, by combining descriptions of the burner acoustic eld with that of the combustion response [START_REF] Dowling | Acoustic Analysis of Gas Turbine Combustors[END_REF][START_REF] Sattelmayer | Assessment of methods for the computation of the linear stability of combustors[END_REF][START_REF] Nicoud | Acoustic Modes in Combustors with Complex Impedances and Multidimensional Active Flames[END_REF]Noiray et al., 2008) even in systems featuring multiple burners such as annular combustors [START_REF] Laera | Flame Describing Function analysis of spinning and standing modes in an annular combustor and comparison with experiments[END_REF] and in real annular engines [START_REF] Bothien | Analysis of References Azimuthal Thermo-acoustic Modes in Annular Gas Turbine Combustion Chambers[END_REF].

The main issue in these dierent descriptions of the combustor dynamics remains that of providing a proper representation of the response of the ame to acoustic waves.

Flame Transfer Functions

In many low-order models used to predict stability maps of combustion systems, the ame frequency response to acoustic waves can be described by a ame transfer function FTF. If one restricts the analysis to fully premixed systems at constant equivalence ratio, the FTF denes as [START_REF] Ducruix | Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms[END_REF]:

F(f ) = Q / Q u /u = G(f ) exp(iϕ(f )) (3)
The FTF relates the heat release rate uctuations to incoming acoustic modulations, as function of the excitation frequency f . It is often expressed as a complex number, in terms of a gain G and a phase dierence ϕ. In the case of complete premixed combustion with constant mixture equivalence ratio, the ame surface A and light emission I are directly proportional to the heat release rate [START_REF] Hurle | Sound emission from open turbulent premixed ames[END_REF][START_REF] Schuller | Dynamics of and noise radiated by a perturbed impinging premixed jet ame[END_REF] and one may write [START_REF] Ducruix | Theoretical and experimental determinations of the transfer function of a laminar premixed ame[END_REF]:

F(f ) = A /A u /u = I /I u /u = G(f ) exp(iϕ(f )) (4)
The FTF of swirling ames can be measured experimentally [START_REF] Kulsheimer | Combustion Dynamics of Turbulent Swirling Flames[END_REF]Kim et al., 2010a) or obtained numerically through Large Eddy Simulations (LES) [START_REF] Freitag | Investigation of a strongly swirled unconned premixed ame using les[END_REF][START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Merk et al., 2018a). Attempts were also made to theoretically model this response [START_REF] Borghesi | Dynamic response of turbulent swirling ames to acoustic perturbations[END_REF][START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF][START_REF] Han | The response of stratied swirling ames to acoustic forcing: Experiments and comparison to model[END_REF].

In this work, we will concentrate more on experimentally determined FTFs.

From a practical point of view, following Eq. ( 4), it is possible to measure FTFs by determining the mean I and uctuating I ame luminosity, by gathering light from the whole combustion region and then comparing these signals with a reference velocity (see [START_REF] Durox | Experimental analysis of nonlinear ame transfer functions for dierent ame geometries[END_REF].

Once the FTF is determined, its combination with a description of the system acoustics let one determine eigenmodes and growth rates of an eventual instability [START_REF] Dowling | The calculation of thermoacoustic oscillations[END_REF]. This linear analyses is unfortunately restricted to small perturbations and cannot account for the limit cycle oscillations observed in many experiments. To overcome this issue, one has to substitute the FTF with its non-linear extension, the ame describing function FDF [START_REF] Dowling | A kinematic model of a ducted ame[END_REF]. In this new expression, the gain G and phase lag ϕ depend on both the frequency f and the amplitude |u | of the perturbation (Noiray et al., 2008):

F(f, |u |) = Q / Q u /u = G(f, |u |) exp(iϕ(f, |u |)) (5)
Examples of experimentally determined FDFs are reported in Fig. 6 for a nonswirling V-ame (left, [START_REF] Durox | Combustion dynamics of inverted conical ames[END_REF] and for a swirling ame (right, Palies et al., 2010a). It is rst evident that non linearities appear in both cases as the amplitude level |u | increases. For the laminar V-ame in Fig. 6-left, it is even dicult to dene a linear regime for the ame response [START_REF] Schuller | A unied model for the prediction of laminar ame transfer functions: comparisons between conical and V-ame dynamics[END_REF]. One also note that for these fully premixed systems, nonlinearities are restricted to the gain of the FDF [START_REF] Candel | Dynamics of Swirling Flames[END_REF]. As the forcing level increases, the gain of the FDF decreases due to saturating mechanisms [START_REF] Lee | Experimental Diagnostics for the Study of Combustion Instabilities in Lean Premixed Combustors[END_REF][START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF][START_REF] Bellows | Nonlinear Flame Transfer Function Characteristics in a Swirl-Stabilized Combustor[END_REF]Oberleithner et al., 2015). On the other side, the phase dierence between velocity and light intensity signals evolves almost linearly with frequency, independently of the forcing level.

An important dierence between non-swirling and swirling ames, is evident from the comparison of FDF shapes. The FDF gain of non-swirling V-ames behaves as a low-pass lter, with a large frequency range over which the gain exceeds unity [START_REF] Schuller | A unied model for the prediction of laminar ame transfer functions: comparisons between conical and V-ame dynamics[END_REF]. It then regularly decreases towards zero for increasing frequencies. The FDF of swirling ames is characterized by an alternating behavior of the gain with maximum and minimum values and by an inection point in the phase lag curve around the frequency where the gain features a minimum value (f = 60 Hz in Fig. 6-right). It is also observed, that the non linearity is strongest at the FDF gain maximum frequencies, while it is almost suppressed at the gain minimum frequency for swirling ames. These dierent responses are the consequence of dierent mechanisms acting for non-swirling and swirling ames. It is worth remembering that both the non swirling ame in Fig. 6-left and the swirling ame in Fig. 6-right are anchored on a central insert in the burner.

Overview of the main mechanisms controlling the ame response to acoustic perturbations Flame transfer functions FTF and ame describing functions FDF are very powerful concepts for engineers to determine the system dynamics and the trajectories of unstable modes, but they do not give access to any qualitative information about the physical mechanisms resulting in this ame response.

An improved comprehension of the mechanisms of unstable combustion in lean premixed burners is still needed for the development of robust systems. To accomplish this task, FTF/FDF measurements should be completed by supplementary diagnostics or numerical simulations analyzing the ame motion or, when possible, by proper modeling of the ame response. Several research groups have put an eort into this direction. In the following, a brief overview of the main mechanisms altering the ame response to acoustic perturbations is presented. We consider only results referring to fully premixed systems, without considering uctuations of the equivalence ratio [START_REF] Lieuwen | Combustion Instabilities: Basic Con-References[END_REF], and where only one ame is stabilized in the combustion chamber, like those studied in this work.

Non-linear ame-vortex interactions

One of the rst experimental eorts, dedicated to the characterization of the non-linear response of swirl-stabilized ames to ow perturbations and to the exploration of the underlying mechanisms for saturation, is that of [START_REF] Bellows | Flame transfer function saturation mechanisms in a swirlstabilized combustor[END_REF] They performed experiments on an atmospheric, swirl-stabilized burner, sketched in Fig. 7. The acoustic oscillations were provided by two loudspeakers mounted in the inlet section. Acoustic velocity disturbances were deduced using the two microphone method [START_REF] Abom | Error analysis of two-microphone measurements in ducts with ow[END_REF]. The relative magnitude of the combustion heat release rate oscillations were obtained from global CH* chemiluminescence measurements, with a photomultiplier tted with a 10 nm bandwidth lter centered at 430 nm. OH planar laser-induced uorescence (PLIF), was also used to obtain instantaneous twodimensional slices of the ame and analyze the ame motion in an axial plane of the burner. [START_REF] Bellows | Flame transfer function saturation mechanisms in a swirlstabilized combustor[END_REF] don't provide full data for the FTF in their measurements. They chose to focus on two conditions leading to a non-linear ame response.

For a forcing frequency f = 130 Hz, they observed a linear behavior of the ame response up to a forcing amplitude u /u 0 of nearly 100%, as shown in Fig. 8left. A signicant amplitude dependence of the phase-lag curve between the u' and CH*' signals was in this case also observed. Saturation takes place at a much lower amplitude for u /u 0 0.35, when the ow is forced at f = 410 Hz. Figure 8-right shows that, in this second case, the phase-lag is almost independent of the forcing level. Figure 8 thus indicates that the nonlinear dynamics of swirled ames could be very complex.

The mechanisms leading to this ame response were analyzed with OH PLIF images of the ame motion at the two forcing frequencies. Results are reported in Fig. 9. At a frequency of f = 130 Hz, they identied ame-vortex interaction as a main mechanism for non-linearity. The consequent roll-up of the ame causes a very rapid reduction in its area and, therefore, in heat release rate, as already demonstrated for laminar non swirling V-ames [START_REF] Durox | Combustion dynamics of inverted conical ames[END_REF]. The ame heat release response does not increase proportionally with the velocity perturbation amplitude, and saturation occurs. The same type of mechanism was observed for non-swirling ames, as previously reported by [START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF] for turbulent V-ames and [START_REF] Durox | Combustion dynamics of inverted conical ames[END_REF] for laminar V-ames.

A dierent mechanism was identied at a forcing frequency f = 410 Hz. At this frequency, non-linearities were triggered by an oscillation of the internal recirculation region, causing an unsteady lifto of the ame, with a consequent reduction of its area (Fig. 9-bottom).

A similar analysis was conducted by Paschereit's group at TU Berlin. The atmospheric combustor test rig used for their investigation is shown schematically in Fig. 10 and fully described in [START_REF] Schimek | Experimental investigation of the inuence of high amplitude forcing and swirl uctuations on the ow eld and the transfer function of a swirl-stabilized ame[END_REF]. The combustor was run in a perfectly premixed mode, with a xed equivalence ratio of φ = 0.70. FDF measurements are presented for forcing frequencies between 50 and 400 Hz in Oberleithner et al., 2015 and reported in Fig. 11. The acoustic forcing was provided by two loudspeakers. The OH*-chemiluminescence was in this case preferred to determine the heat release rate. It was measured by a photomultiplier with a 307 nm band-pass lter. The velocity uctuation was measured with a 2-component LDV system (Fig. 10). The FDF displayed in Fig. 11 shows a strong non-linear behavior with respect to the forcing amplitude. The saturation of the ame response is most emphasized around the peak frequency of the linear (or low amplitude forcing) FTF, indicating that the mechanism that causes this peak becomes increasingly less eective with higher forcing amplitudes. The FTF phase-lag in Fig. 11-b, is here again found to be almost independent of the forcing level and conrms many other experiments (Palies et al., 2011c). The fact that a dierent behavior is observed in the study of [START_REF] Bellows | Flame transfer function saturation mechanisms in a swirlstabilized combustor[END_REF] might be due to the use of the CH* radical, which is more sensitive to soot observation than OH* emission, to detect heat release rate uctuations.

The saturation process was further investigated by forcing the ame at dierent amplitudes at the constant frequency of f = 254 Hz and by OH* imaging and PIV measurements of the ow and ame responses. Figure 12 shows results at dierent forcing amplitudes. An isocontour of the Q criterion is superimposed onto the phase-averaged OH* images highlighting the development of coherent ow structures during one forcing cycle. A vortex ring is generated at the outer edge of the burner mouth, it propagates along the outer shear layer, it impinges on the ame and roll-up its tip. As Bellows et al., these authors identify amevortex interaction as one of the leading mechanism for the non-linear response of swirling ames.

Swirl uctuations

Another important mechanism, which is at the origin of the dierences observed in Fig. 6 between the responses of non-swirling and swirling ames, is the interference between acoustic and azimuthal convective perturbations.

Let us consider the schematic in Fig. 13. When an acoustic wave impinges on a swirler, on the downstream side of the swirler, one still nds a transmitted acoustic wave, but also a generated vorticity wave. These two waves travel at a dierent speed, the former being acoustic in nature and thus travelling at the local speed of sound, the latter being convective in nature and thus travelling at the local mean ow speed. The interaction between these acoustic and convective perturbations eventually give rise to swirl number oscillations (Palies et al., 2011a).

Komarek and Polifke, 2010 specically designed an experimental setup to analyze the impact of uctuations of the swirl number on the heat release rate of a fully premixed ame. Figure 14 shows a sketch of the experimental setup they used. The burner featured an axial swirl generator with variable axial position. It was then possible to change the relative phase of the swirl uctuations and to study the impact on the dynamic ame response. By increasing the distance of the swirler to the ame, the time required for convective transport of a uctuation in swirl number to the ame was also increased.

Their FTF data are reported in Fig. 15. The excitation was realized by a siren that modulated the ow rate of the incoming mixture to the plenum. The velocity signal was obtained by a Constant Temperature Anemometer and the heat release signal was deduced with a photomultiplier in combination with an interference lter to determine the OH* emission. When measuring the FTF for dierent positions of the swirl generator, a strong impact on the response of the ame was observed. The ame response is characterized by a series of local minimum and maximum values in Fig. 15. With increasing time lag (increasing distance between the swirler and the ame), the prominent local minimum of the FTF gain moved to lower frequencies. For the swirler mounted closer to the ame (∆x = 30 mm in Fig. 15) only a local maximum of the ame response was observed, but it is likely that a local minimum would appear at an higher frequency.

An important outcome of the analysis of [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF], is that the interaction between acoustic and convective waves, leading to swirl uctuations, is responsible for the shift in the frequency range of the FTF gain curve. The next step was to investigate the conditions leading to a maximum or a minimum ame response. The current interpretation follows the analysis described by Palies and coworkers in a series of articles based on experimental, numerical and theoretical work (Palies et al., 2010b;Palies et al., 2011e;[START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF]. Their interpretation is based on the constructive/destructive interference of swirl number uctuations and vortex shedding. The experimental setup used for their analyses is sketched in Fig. 16. These interpretations have been later questioned by [START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. The experimental setup used for their analysis is sketched in Fig. 20. The burner was fed by mixtures of natural gas and air for dierent bulk velocities and equivalence ratios. They performed several FTF measurements and observed a qualitative similarity of the response for all the operating conditions. An They further analyzed the ame response mechanism with the same windowing method applied by Palies et al., 2010b. First, they investigated the sensitivity of the results upon the window division location, and regardless of this choice, constructive interference between the heat release uctuations in the upper and lower regions of the ame could not be seen.

The envelope of the mean ame position uctuations was determined at the ame transfer function gain extrema frequencies and is shown in Fig. 22. They found that, at gain minima frequencies, uctuations in mean ame position near the ame base were present, indicating the presence of swirl number uctuations at these frequencies. In contrast, uctuations in the mean ame position near the ame base were much lower at the gain maximum frequency (middle image in Fig. 22).

These results from Bunce et al. conrm those from Palies et al. in terms of destructive interference between ame angle uctuations and vortex rollup at the minimum gain frequency, but question the constructive interference at the frequency corresponding to a FTF gain maximum, since ame angle uctuations were seen not to exist at that particular frequency in the measurements of [START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF].

Upstream and downstream forcing

In all the examples shown until now, the FTF or FDF ware determined by introducing acoustic perturbations coming from the upstream side of the ame.

In practical systems, the interference between acoustic waves and the ame can happen from both the upstream and downstream sides. It is then interesting to compare FTF results obtained by submitting the ame to perturbations coming from its downstream side (Hochgreb et al., 2013;[START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF].

We focus here on the results from [START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF] The experimental setup used for their analysis is sketched in Fig. 23. The burner is fed by a mixture of methane and air at xed equivalence ratio (φ = 0.82) and bulk velocity (U b = 5.4 m/s). In these experiments, the ame is forced either by a loudspeaker connected to the bottom of the burner in the fresh gas region or by a set of loudspeakers connected to the combustion chamber exhaust tube in the burnt gas region.

They compared several FDFs with dierent reference signals. The results reported in Fig. 24 were obtained by measuring the velocity uctuations with a hot wire probe (HW in Fig. 23) located below the swirler and the heat release rate uctuation with a photomultiplier equipped with an OH* lter. At rst it was shown that the FDFs measured with the two dierent forcing methods, shown in Fig. 24, follow the same trends. The peculiar shape of swirling ame FDFs is retrieved in both cases and maximum and minimum values of the gain are the same and appear at the same frequencies. Also the non-linear behavior for the gain and linear behavior for the phase was conrmed.

The experimental campaign was followed by a theoretical analysis. From Fig. 24, one can see that slight dierences are observed in the two FDF gain 0.10 0.20 0.30 0.40 0.55 0.72

Figure 24: Flame describing functions obtained with upstream (left) or downstream forcing (right) for six dierent forcing levels u /u at the hot wire location. Reproduced from [START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF] curves. This was attributed to the fact that the upstream and downstream acoustic boundary conditions are modied depending on the type of forcing used to excite the ame, which in turn modies the acoustic elds inside the system. In these experiments, the forcing level u /u is xed at the hot wire location, for both upstream and downstream excitation. Thus, it does not correspond to the forcing level seen by the ame at the injector outlet. It was shown that a large acoustic pressure drop, due to acoustic losses, is generated across the swirler in the case of upstream forcing. This acoustic pressure is much lower in the case of downstream forcing. By taking into account this acoustic pressure drop to reconstruct the acoustic velocity at the burner outlet, [START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF] were able to obtain an almost perfect match between describing functions measured with upstream or downstream forcing at all frequencies and forcing levels. This proves that the mechanisms acting on the ame to determine its response are the same, regardless of the direction of incoming acoustic waves. When the correct velocity signal at the injector outlet is used to determine the FTF/FDF, quantitative results are also independent of the direction of incoming acoustic waves. It is however dicult in many systems to measure this signal directly at the burner outlet and the FTF/FDF are often determined with a reference signal away from the injector outlet. In these cases, reconstruction rules of the FTF at the burner outlet, presented in [START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF], should be used.

Aerodynamically stabilized ames

For all the previously presented cases, a central blu body was used to help ame stabilization. Flame transfer function measurements, for swirling ames fully aerodynamically stabilized away from all solid components, as in many real engines, are less well documented. One example of such measurements is provided by [START_REF] Durox | Flame dynamics of a variable swirl number system and instability control[END_REF]. They performed tests on the burner sketched in Fig. 25, with a methane-air mixture, with φ = 0.70 and U b = 11 m/s. A radial swirler, with variable blade angles, is used for ame stabilization. When the blade angle is varied, striking changes in the ame conguration were observed, as shown in Fig. 26. When the swirl number takes large values (i.e., for high blade angles), a characteristic swirling ame shape with a conical root lying on the burner axis, was obtained. Gradually, as the angle decreases the ame was positioned further downstream in the combustion chamber. The FTF is in turn strongly aected by a change in blade angle, as reported in Fig. 27. In this study, the velocity uctuation is measured by LDV at the burner outlet and the heat release rate deduced from the OH* signal measured by a photomultiplier gathering light from the whole ame. The gain features large values up to 200 Hz, followed by a large drop, regardless of the blade angle. Except for results presented for a blade angle of 61 o , one cannot distinguish a clear drop of the FTF gain at low forcing frequencies. The phase of the FTF increases almost linearly with frequency, for all the blade angles, with a phase lag that regularly increases as the blade angle drops. No attempt was made to investigate the mechanisms controlling the response of these aerodynamically stabilized ames.

Thesis objectives and contents

This brief literature review indicates that the mechanisms determining the response of premixed swirled ames to acoustic perturbations are still not fully understood. In particular, eects of the swirl level, conditions leading to a drop of the FTF gain and eects of the central insert remain to be elucidated. This thesis is dedicated to the study of the response of lean premixed swirling ames to acoustic perturbations, for dierent congurations of the swirling injector. A laboratory-scale model gas turbine combustor, powered by turbulent premixed ames at dierent lean injection conditions and operated at atmospheric pressure, is adapted from a previous setup for this purpose.

The problem is tackled from a fundamental point of view, analyzing the elementary processes associated to changes of the injector design, leading to a reduction of the ame response to acoustic perturbations in a simplied conguration, portraying the combustion dynamics of real systems. This manuscript is made of nine chapters, which content is briey described in the following.

Chapter 1 to 3: Burner details and applied methodologies

This rst part is dedicated to the description of the burner geometry and of the main diagnostics applied in the process of this work. Chapter 1 rst focuses on the experimental setup, which is there described in detail. As this work is mainly dedicated to the assessment of the eects of dierent swirling injection conditions on the mean and dynamic properties of lean-premixed swirling ames, a burner with some easily replaceable components was designed. In chapter 1 details of the dierent injectors developed in this study are given. Chapter 2 is dedicated to a rapid description of the main acoustical and optical diagnostics used in this study, with also some examples of measurements.

First, the technique used to measure FTFs is presented. The ow elds properties obtained for dierent injection conditions are analyzed through Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV) measurements. Some examples of ame shapes under steady and forced conditions obtained with an ICCD camera are then presented. To close this rst part, chapter 3 presents the numerical solver and the meshed geometry used for the Large Eddy Simulation (LES) analysis that was carried out in conjunction with the experiments.

Chapter 4 and 5: Inuence of the injection conditions on the mean ow and ame properties and on thermoacoustic instabilities

The second part of the manuscript focuses on the description of the results obtained for the dierent swirling injection conditions tested. Chapter 4 present ow and ame properties. The investigated geometry modications cover effects of radial swirler, injector and blu-body main dimensions. Also the mixture ow rate is modied. A stabilization chart of the burner is shown to be divided into three regions separating the blowo regime from the ashback regime and well-stabilized ames. Flow elds properties in cold ow conditions and ame shapes in reactive conditions, are then analyzed for the dierent injectors tested. Chapter 5 reports measurements when the system features combustion instabilities. These instabilities are triggered by modifying the length of the exhaust tube. Eects of modications of the swirler design and of the distance between the swirler and the combustion chamber backplane are examined.

Chapter 6: Inuence of the injection conditions on the transfer function of lean-premixed swirling ames

The third part of the manuscript, covered in chapter 6, is dedicated to the presentation of FTF/FDF measurements obtained by submitting ames to acoustic perturbations, generated by a loudspeaker mounted on the upstream side of the ame. Eects of dierent injection conditions are considered. The typical shape of the FDF of swirling ame is retrieved. All the geometrical congurations tested conrmed that a decrease of the minimum gain of the FTF can be obtained by an increase of the swirl number S. A limit is reached when an excessive swirl number leads to ashback of the ame into the injector. For a xed swirl level S and modifying the mean ow velocity or the swirler to Introduction chamber backplane distance, the FTF curve gets shifted in the frequency range without modications of the gain level.

Chapter 7 to 9: Mechanisms controlling the FTF shape

This last part is dedicated to the interpretation of results presented in chapters 5 and 6. In chapter 7, phase locked ICCD and PIV measurements and LES are carried out at peculiar frequencies deduced from the FTF results of chapter 6.

It is rst shown that the motion of the ame is quantitatively but not qualitatively dierent at frequencies corresponding to a minimum or a maximum gain value. Strong swirl number uctuations are observed at both frequencies, in simulations as well as in experiments. The ame response is dominated by the interaction with vortical structures generated at the burner rim. The strength of these structures is attenuated at the minimum FTF gain frequency, even in cold ow conditions. This is attributed to the dierent phase shift between acoustic and convective perturbations travelling from the swirler exit to the ame, that eventually lead to a dierent velocity prole at the chamber backplane, resulting in dierent strength of vortical structures generated there. Chapter 8 then proposes a scaling of the FTF results by properly accounting for the dierent time scales of (i) the convection of azimuthal velocity perturbations from the swirler exit to the chamber backplane and (ii) the convection of vortical structures from the burner rim to the ame. Chapter 9 nally present the dierences in terms of ame response mechanisms, when the ame stabilization mechanism is modied. In particular, three cases are considered: (i) a non-swirling ame stabilized behind a blu-body, (ii) a swirling ame stabilized with the help of the same blu-body and (iii) an aerodynamically stabilized ame without blu-body. In these two last cases, the swirl is kept roughly to the same level. It is shown that the response of the aerodynamically stabilized ame shows a low-pass lter behavior, without any reduction of the gain in the intermediate frequency range. Surprisingly, its response is closer to that of a non-swirling ame. Three competing mechanisms denes the dynamics of these ames: ame-vortex interaction, ame angle oscillations and vertical oscillations of the ame root position when the blu-body is removed. The dominant one depends upon the ame stabilization mechanism. The reduction of ame response to acoustic waves in the intermediate frequency range, possibly reducing the sensitivity of LPM systems to dynamic phenomena, could only be observed for the conguration where ames are stabilized thanks to a swirling stream and with the help of a blu-body.

Experimental setup

The NoiseDyn setup has been designed with the objective of exploiting it for various type of combustion analyses. Due to its versatility, it has been used, for instance, for the analysis of laminar conical ames dynamics, for the analysis of swirling ames dynamics and also for the analysis of combustion noise. In this work, only the components needed for the analysis of the dynamics of swirling ames are presented.

Information about other analyses conducted with this setup can be found in the Ph.D. thesis manuscripts from R. Gaudron [START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF] and M. Merk (Merk, 2018). This chapter introduces the geometrical details of the burner. A general description is given in Section 1.1, followed by a more detailed characterization of some replaceable components in Section 1.2.

Burner geometry

The setup used for the analyses conducted in this study is schematically sketched in Fig. 1.1, also showing the air and methane supply lines. A picture of the burner, together with the design drawn with the CAD software Autodesk Inventor, are shown in Fig. 1.2. Fully premixed methane and air are injected from two diametrically opposed apertures positioned at the burner base. The air and the methane are supplied separately to a mixing box, at ambient temperature, by a series of pipes connected to an air compressor and to a pressurized gas tank, located outside the experimental room. The mass ow rates of methane and air injected are controlled by Bronkhorst F-Series mass ow controllers. The mixing box is located far upstream the burner bottom apertures, in order to assure a complete premixing of the reactants before their injection in the burner.

When the mixture enters at the bottom of the burner in a 65 mm diameter cylindrical tube, it rst crosses a perforated grid, to homogenize the ow and prevent the risk of ame ashback into the pipes and an honeycomb structure that breaks the largest turbulent scales. It then ows through an upstream plenum of 65 mm diameter featuring some wall pressure taps for acoustic pressure measurements. These and the other internal components of the burner are highlighted in Fig. 1.3, showing a longitudinal cut of the geometry through its central axis.

The mixture then travels along the upstream plenum to reach a rst convergent unit (contraction ratio: 8.73). At the outlet of the nozzle of 22 mm diameter, a top-hat velocity prole is obtained. The values of the bulk ow velocity U b presented all along this work, will always be referred to this section ( = 22 mm, T u = 293 K, p u = 1 atm). At the same axial location, a microphone and a hot wire anemometer probe are used to measure the local ow velocity outside the boundary layer and the acoustic pressure. An enlargement of the passage section is then necessary to mount the swirling device. More details about the geometry of the swirler units are given in Section 1.2. The swirler unit imparts a certain amount of rotation to the ow. On the downstream side of the swirler, up to the combustion chamber, the injector geometry is highly adjustable and many dierent congurations were tested during this work, as detailed in Section 1.2. The temperature of the top of the burner is kept as close as possible to the ambient value by water cooling. The mixture then exhaust in the combustion chamber through an enlargement of the cross section.

The squared combustion chamber is made of four transparent quartz windows to allow optical access. The four windows are kept in position by four steel rods at each corner of the chamber. Between the rods and the quartz, high temperature seals exclude the risk of air leakage or external air intrusion. A high-voltage spark igniter (7 kV and 30 mA), inserted through a small hole in one of the steel rods on the corner of the chamber, is used to initiate combustion. A second convergent unit (contraction ratio: 1.27), changing the cross section from squared to circular, is placed on top of the combustion chamber to guide the hot combustion products out of the burner into the atmosphere. The burner is operated at atmospheric pressure.

The geometry of the bottom and top parts of the burner, before the upstream plenum and after the exhaust convergent unit, is modied according to the type of analysis to be conducted. For ame dynamics analysis under external acoustic excitation, a loudspeaker (Monacor SP-6/108PRO, 100 Watts RMS) is mounted at the bottom of the burner and used to introduce acoustic excitations of dierent frequency and amplitudes in the ow. For measurements in the absence of external forcing or thermo-acoustic instabilities characterization, the bottom side of the burner is closed by a plate equipped with a microphone, as shown in Fig. 1.3, while exhaust pipes of dierent lengths can be mounted downstream the second convergent unit. More details about these components are given in the following.

Burner replaceable components

One of the most important characteristics of the NoiseDyn setup is its versatility. Many of the burner components can be demounted and substituted or simply removed from the test-rig, to test the eects of dierent burner geometries on the combustion properties. In some cases, components are substituted to perform dierent type of analyses, as for the measurements with or without external acoustic excitation. The following sections of this chapter are dedicated to the description of the main burner replaceable components.

Swirlers

The swirlers used for the experiments conducted in this study are radial vanes with tangential inlets. A total of ve dierent swirlers were designed. A picture of one of these components is shown in Fig. 1.4 together with a sketch of two representative designs and a table reporting the parameters modied in the ve dierent geometries tested. The number n of the tangential inlets, the diameter D s of these tangential inlets and the semi-distance H between the axes of two diametrically opposed inlets (Fig. 1.4), are modied to select the rotation rate imparted to the ow. The design of swirlers SW 0 and SW 3 only diers by the distance H in Fig. 1.4. In the rst device, designated as SW 0 , the swirling vanes are aligned with the radial direction (H = 0 mm). In the second one, designated as SW 3 , the vanes are shifted from the radial direction by H = 6 mm, to impart a strong rotation to the ow. The values of H, D s and n for the other swirlers tested in this study are reported in the table in Neglecting the pressure term in the axial momentum ux [START_REF] Beer | Combustion Aerodynamics[END_REF], the swirl number S is expressed in the form:

S = 2 D o u θ u z r 2 dr u 2 z rdr (1.1)
The swirl number should, when possible, be determined from measured values of the axial, u z , and azimuthal, u θ , velocity proles, but this is obviously not possible during the design stage of a swirler. In this case, S can be estimated with analytical expressions, taking into account the swirler and injector geometries and making assumptions about the axial and azimuthal velocity components.

One should however keep in mind that such estimates can notably dier from values deduced from experiments [START_REF] Palies | Dynamique et instabilites de combustion des ammes swirlees[END_REF][START_REF] Durox | Flame dynamics of a variable swirl number system and instability control[END_REF]. From Fig. 1.5, a mass balance yields the ratio between the axial velocity ūz and the velocity inside the swirler injection channels ūh :

ūh ūz = 1 n R 2 o -R 2 i R 2 s (1.2)
where

R s = D s /2, R o = D o /2 and R i = D i /2 = 3
mm is the radius of the central rod, shown in Fig. 1.3-a. It is then straightforward, from trigonometric considerations, to get the ratio between axial and azimuthal velocities:

ūθ ūz = sin α n R 2 o -R 2 i R 2 s (1.3)
where sin α = H/R o .

To make estimates of the swirl number, one can further make assumptions about the axial and azimuthal velocity components. One possibility is to assume a uniform axial ow prole u z (r) = ūz and a solid body rotation prole for the U U h U r showing the azimuthal velocity component, the ūθ prole reaches slightly higher values for SW 3 than for SW 1 . This is due to the higher value of the distance H between two diametrically opposed swirler inlets, leading to a higher rotation rate.

S sin α 2n R 4 o -R 4 i R 2 s R 2 o (1.
The swirl numbers SW 3 = 0.73 and SW 1 = 0.63 are deduced from Eq. (1.1) and are superposed in Fig. 1.6 on top of the analytical estimates. One can see that Eq. (1.4), considering a constant axial and a linear azimuthal ow component, leads to an underestimation of the experimental swirl number. As a matter of fact, the axial velocity prole shown in Fig. 1.7 is not uniform along the distance going from the central rod at R i = 6 mm to the injector rim at R o = 10 mm. In this case, a better prediction of the swirl number could be obtained by considering a linear axial velocity evolution u z (r) = ūz (r/R o ) and a solid body rotation for the azimuthal velocity prole u θ (r) = ūθ (r/R o ). With this new assumption, integration of Eq. (1.1) leads to:

S 4 5 sin α n 1 R o R 2 s R 5 o -R 5 i R 2 s + R 2 o (1.5)
Swirl estimates obtained with these new approximations are reported as dashed )) for swirlers SW 1 (white circles) and SW 3 (black diamonds). Solid lines: estimates obtained with a uniform axial velocity prole and a linear azimuthal velocity prole. Dashed dotted lines: estimates with linear axial and azimuthal velocity proles (Eq. (1.5)). Superposed as blue (SW 1 ) and red (SW 3 ) crosses, the swirl numbers measured by LDV. dotted lines in Fig. 1.6. They show a better agreement with the experimentally determined values. This later formula is retained to estimate the swirl numbers in this work when they were not determined from LDV or PIV measurements.

A change of the geometrical parameters of the swirler does not only modify the rotation rate of the ow, but also the pressure drop across the swirler unit. This loss was measured for the dierent swirler designs between the upstream plenum and the atmosphere with a liquid column U-tube manometer. One of the port of the manometer was connected to the pressure tap in the upstream plenum closest to the swirler (Fig.

Central injection unit

The ow exits the swirler into a central injection tube (Fig. 1.9-a). The injector geometry, from downstream the swirler up to the combustion chamber backplane, is highly adjustable. The diameter of this tube is D = 22 mm, over a rst section of variable length δ 1 , followed by a central insert of length 34 mm, and diameter D o . The outer diameter D o of the injector can be modied by substituting this central insert, shown in Fig. 1.10-a. A central rod of diameter D i = 6 mm, ending with a cone of variable diameter at the top C = 10, 12 or 14 mm (Fig. 1.10-b), is inserted in the injection tube to ease stabilization of the ame. The distance between the swirler exit and the combustion chamber backplane can also be modied by 5 mm increments: δ = 35, 40, 45 or 50 mm using a set of spacers of height 5 mm shown in Fig. 1.10-c. To modify the distance δ, one can use from one to four spacers. This conguration with a central rod in Fig. 1.9-a is the one which is retained all along this study, except in chapter 9 where the injector in Fig. 1.9-b is used to analyze the response of fully aerodynamically stabilized ames. In this second case, the central insert with diameter D o = 12 mm is terminated by a diverging cup with an half angle β = 15 • (Fig. 1.9-b).

Conclusions

The setup developed in this study allows studying the dynamics of swirling ames with and without acoustic forcing for a wide set of injector congurations. Two main classes can be dened giving that the injector features or does not feature a central insert to help stabilizing the swirled ames. The injector without insert is used in chapter 9 to analyze the dynamics of fully aerodynamically stabilized ames. The other injectors with a central insert are used in the rest of this work. A parametric analysis has been conducted with this second class of injectors for which 28 congurations, indicated in table 1.1, were tested by changing the swirling vane (SW 0 , SW 1 , SW 2 , SW 3 , SW 4 ), the injector outlet diameter varying from D o = 18 to 22 mm, the diameter of the conical end piece C = 10 to 14 mm, the distance δ between the swirler and the injector outlet from 35 to 40 mm and the bulk ow velocity from U b = 2.72 to 5.44 m/s. The resulting ow was characterized by LDV or PIV under cold ow conditions in 12 congurations. These data were used to determine the swirl number S. In the 16 other cases the swirl number was deduced with analytic approximations (Eq. (1.5)).

Diagnostics

The main measurement techniques applied within the scope of this work are introduced in this chapter. Techniques used to force the ame and determine the velocity and heat release rate uctuations for ame transfer function (FTF) calculations are described in the rst part. A constant temperature anemometer probe (HW) and a photomultiplier (PM)

equipped with an OH* lter are used for this purpose. In Sec. 2.2, Laser Doppler Velocimetry (LDV) used to measure axial, azimuthal and radial velocity components, as close as possible to the injector exit, is briey described. Particle Image Velocimetry (PIV), adopted for the characterization of velocity elds in dierent planes inside the combustion chamber in steady and pulsated conditions, is presented in Sec. 2.3. For steady conditions, PIV measurements are performed in cold and reactive conditions. Lastly, Sec. 2.4 describes the optical arrangement used to characterize the steady and forced ame structures.

Flame transfer function measurements

A consistent part of the work conducted in this thesis is dedicated to analyze the eects of geometrical modications of some injector components on the ame response to incoming acoustic perturbations. This response is characterized by measuring the ame transfer function or ame describing function linking velocity uctuations at some location upstream the ame and heat release rate uctuations. For this scope, the burner is equipped with a loudspeaker (Monacor SP-6/108PRO, 100 Watts RMS) mounted at the bottom of the burner.

A homemade LabVIEW routine generates a digital harmonic signal, which is converted to an analog signal and sent rst to an amplier and then to the loudspeaker. The forcing signal is generated at dierent amplitudes and frequencies. The excitation level produced by the loudspeaker is controlled by a hot wire (HW in Fig. 2.1). The hot wire measurement is then fed back to the routine and the forcing signal amplitude is adjusted until the desired u /ū level is obtained, where u denotes the root-mean-square value and ū the av- in Fig. 2.2 are also very close, indicating that the LabVIEW routine controlling the forcing level behaves well. One may conclude that the velocity modulation obeys to: u(t) = ū + u √ 2sin(ωt). The calibration of the hot wire probe was carried out with a mixture of pure air. It was shown by [START_REF] Guiberti | Analysis of the topology of conned premixed swirling ames[END_REF] that the response of the hot wire probe is nearly superimposed for mixtures of methane/air or for pure air mixtures, since the kinetic viscosity and the thermal conductivity of methane and air are close.

OH* chemiluminescence

For perfectly premixed conditions like those studied in this work, the OH* radical emission of the ame is a good measure of its heat release rate. This has also been veried for swirling premixed ames (Palies et al., 2010b) for which it has been checked that Q / Q = I OH * / ĪOH * , where I OH * is the light intensity collected. A photomultiplier (Hamamatsu H5784-04) equipped with a bandpass interferometric lter (Asahi Spectra ZBPA310) centered on 310 nm, with a 10 nm bandwidth, corresponding to the OH* emission wavelength [START_REF] Gaydon | The Spectroscopy of Flames[END_REF] In contrast to velocity signals, the heat release rate perturbations strongly dier in the two plots, both in terms of amplitude of perturbation and of phase shift with respect to the velocity signal. This is due to the delayed response of the ame to incoming perturbations [START_REF] Crocco | Aspect of Combustion Stability in Liquid Propellant Rocket Motors Part I[END_REF] and to the nonlinear response of combustion to ow perturbations [START_REF] Bellows | Nonlinear Flame Transfer Function Characteristics in a Swirl-Stabilized Combustor[END_REF][START_REF] Durox | Experimental analysis of nonlinear ame transfer functions for dierent ame geometries[END_REF]. These signals feature a larger harmonic content in the power spectra shown in Fig. 2.2. The main peak remains however at least one order of magnitude larger than the second harmonic. This will be reected in dierent values for the gain and phase of the FTF. It is also evident that signals can be aected by noise. It is then necessary to apply proper signal processing in order to extract useful information from the signals recorded.

Signals post-processing

Once the desired forcing level is applied, the heat release rate and velocity uctuations are recorded simultaneously to obtain the Flame Transfer Function (FTF). When this procedure is repeated for dierent forcing levels, the Flame Describing Function (FDF) is obtained. For all the FTF measurements conducted in this study, the signals from the HW and the PM are recorded over a duration t = 4 s at a sampling frequency f s = 8192 Hz. The Cross Power Spectral Density (CPSD) and the Power Spectral Density (PSD) of these signals are obtained with a modied Welch periodogram method, dividing the signal into at least 8 segments windowed with an Hanning window and using a 50% overlap. This assures that at least 50 periods of oscillation are considered for the average for each measurement, which is very important since signals are affected by noise. Examples of PSD from HW and PM measurements are shown in Fig. 2.2-bottom for forcing frequencies f = 120 Hz (left) and f = 200 Hz (right). As for the velocity power spectra, the heat release rate power spectra show a strong PSD peak at the excitation frequency. A smaller peak is observed at the rst harmonic, at double the forcing frequency. The heat release rate uctuation is stronger at f = 200 Hz than at f = 120 Hz, which reects the complexity of the ame response to low frequency acoustic perturbations. This is best seen by determining the FTF/FDF as:

F DF = S xy S xx = G(ω, u /ū)exp(iϕ(ω, u /ū)) (2.1)
where S xy is the cross-power spectral density (CPSD) between the velocity ū and the heat release rate I OH * signals and S xx is the power spectral density (PSD) of the velocity signal. An example of FTF result is shown in Fig. 2.3. The gain of the FTF is much higher at f = 200 Hz than at f = 120 Hz as could be deduced from Fig. 2.2. At f = 120 Hz the FTF gain is at its minimum value, while it is almost maximum at f = 200 Hz. The phase-lag value are also dierent at the two frequencies. At f = 200 Hz, signals from the HW and the PM are in phase opposition (ϕ = π), while ϕ π/2 at f = 120 Hz.

LDV measurements

A 2-components LDV system, similar to the one used by [START_REF] Providakis | Compétition entre structures aérodynamiques et modes acoustiques dans une amme swirlée : Inuence de la répartition de carburant[END_REF][START_REF] Mirat | Analyse des instabilites de combustion dans des foyers de centrale thermique fonctionnant au oul lourd[END_REF][START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF], is used to measure the axial, azimuthal and radial velocity downstream the outlet of the injector for selected congurations.

The two laser components of wavelengths λ = 514.5 nm (green component, axial velocity) and λ = 488 nm (blue component, azimuthal/radial velocity) are generated by a continuous 3 W Argon laser. Figure 2.4 shows a sketch of the LDV system from lateral (a) and top (b) views. Measurements with two laser components are conducted simultaneously 1.5 mm downstream the conical end piece of the central insert, which means that either the axial and azimuthal or the axial and radial velocities could be recorded at the same time. The burner is displaced along the axis 1 or 2 (Fig. 2.4) by an automatic controller by steps of 0.5 mm in the radial direction. The transmitting optical system is equipped with a Bragg cell at 40 MHz to discriminate negative and positive velocities. The laser leaves this system through an achromatic lens of focal length 250 mm. The distance between the two laser beams is 40 mm at the exit of the transmitting optic. This generates interference fringes spaced of 3.23 µm for λ = 514.5 nm [START_REF] Mirat | Analyse des instabilites de combustion dans des foyers de centrale thermique fonctionnant au oul lourd[END_REF].

The ow is seeded with micrometric oil droplets of 3 µm diameter by a nebulizer fully described by [START_REF] Durox | Flow seeding with an air nebulizer[END_REF]. When these particles cross the measure- ment volume, the scattered Mie signal is collected by the receiving optic of focal length 300 mm, placed at an angle of 30 • with respect to the laser beam, by photomultipliers with narrow-band interference lters centered on the laser wavelengths. A frequency counter is used to deduce the velocity from Doppler bursts resulting from the droplets passage across the interfering region. The data collection rate is always greater than 10000 s -1 and for each measurement point at least 250.10 3 particles are considered, in order to obtain fully converged mean and root-mean-square (rms) values for the dierent components of velocity. The statistical bias is corrected by the transit time of each particle [START_REF] Mirat | Analyse des instabilites de combustion dans des foyers de centrale thermique fonctionnant au oul lourd[END_REF].

An example of LDV measurements is reported in Fig. study are done without connement, meaning that all the burner components downstream the injector exit are removed from the setup. This was necessary to get a good optical access to this region and have a good signal to noise ratio. The receiving optics is placed at an angle of 30 • with respect to the laser beam and a large fraction of the eld of view would be obstructed by the steel rods at the corner of the combustion chamber. LDV measurements are conducted only in cold ow conditions. Measurements performed in cold ow conditions in this study, are conducted with pure air. When these measurements are done to analyze the ow in reactive conditions, the missing mass ow rate of CH 4 is accounted for by increasing the mass ow rate of air, in order to keep a constant volume ow rate and the same bulk ow velocity U b in the 22 mm diameter section at the convergent outlet.

PIV measurements

Particle Image Velocimetry (PIV) is used in this work to analyze the structure of the velocity eld in dierent two-dimensional planes. This technique is widely used in uid-dynamics and combustion and well described in the literature. The reader is referred for instance to the textbook from [START_REF] Rael | Particle Image Velocimetry, a practical guide[END_REF] for further information. In the following, only the specic settings of our setup are described.

To do PIV measurements, it is rst necessary to seed the ow with particles.

In this study, PIV measurements are conducted at steady conditions with and without combustion and also in cold conditions under external acoustic excitation. For measurements without combustion the same oil particles as for LDV measurements are used to seed the ow. These liquid particles could not be used for measurements under reactive conditions since they evaporate through the ame front. In this case, ow seeding is obtained with solid ZrO 2 (Zirconium dioxide) particles of diameter d 3 µm, having a melting temperature of T = 2988 K. The Stokes number of these ZrO 2 and oil particles is small (S t 1), so that the velocity measured can be considered suciently close to the ow velocity.

The basic principles of the technique is described as follows. Two laser pulses, separated by a well controlled time delay ∆t, illuminate the particles in a measuring plane. This plane is made by expanding the laser beam with a cylindrical (f = 30 mm) and a spherical (f = 500 mm) lens in a laser sheet of about 50 mm height and 1.5 mm width. A dual-shot CCD camera takes a picture at each of this two instances. One Mie scattering image for an axial and a transverse plane through the setup is shown in Fig. 2.7 when the cold ow is seeded with oil particles. A Mie scattering image is also shown in Fig. 2.8 for an axial plane with ZrO 2 particles. Figure 2.7 shows that seeding is distributed homogeneously in the cold ow. Figure 2.8 highlights that in the hot region of the ow the number of particles detected drops signicantly due to thermal expansion of the ow through the ame. This leads to relatively low seeding rates in the central region of the ow. These pictures are then post-processed by a specic software. In particular, they are divided into a certain number of overlapping windows, each window having a certain number of particles inside it. The Dantec Dynamics software is used for images acquisition and images correlations to obtain the velocity. The resolution and precision of the measurements depends upon dierent factors, like the particle size, the quality of the seeding, the intensity of the laser sheet and the relation between the selected ∆t and the range of investigated velocities [START_REF] Rael | Particle Image Velocimetry, a practical guide[END_REF].

The 2D-2C PIV system is presented in Fig. 2.6. It features a 2×400 mJ Nd:YAG laser doubled at 532 nm operated at 10 Hz and a 2048×2048 pixels CCD camera (Dantec Dynamics, FlowSense EO 4M). For measurements conducted in reactive conditions, a 10 nm bandpass lter centered at 532 nm was placed in front of the camera lens to avoid detecting light from the ame. A Delay-Pulse generator is used to synchronize the laser pulses and the camera apertures. Two dierent optical setups are used with a time interval between the two laser pulses ∆t = 10 µs and a pixel pitch of 27.88 px/mm for measurements made in an axial plane crossing the burner axis and ∆t = 25 µs and a pixel pitch of 40.14 px/mm for measurements made in a transverse plane at a distance of 2 ± 0.5 mm from the blu-body. In this last case, a mirror placed on top of the combustion chamber and inclined by an angle of 45 • is used to visualize the horizontal laser sheet, which is obtained by a 90 • rotation of the cylindrical lens as shown in Fig. 2.6. In both cases the laser sheet has a length/height of 50 mm and a thickness of about 1.5 mm in the region of interest. This large thickness helps minimizing out of sheet particle displacements of the swirling ow between the two laser pulses. Eight hundred images are taken to assure the convergence of the mean and rms values of the velocity eld, which is computed from the cross-correlation of the PIV images by a three passes window deformation technique (from 64×64 pixels to 16×16 pixels interrogation areas), with an uncertainty of 0.1 pixel on Examples of measurements for the mean and root mean square RMS axial component of velocity are reported in Fig. 2.9 in steady cold (left) and hot (right) conditions. These measurements are conducted in the central axial plane. One can clearly observe the recirculation zone with negative velocities in the central region downstream the blu-body, typical of swirling ows. This region is larger and longer in the right of Fig. 2.9 for reactive conditions. One can also identify the jets of high velocities at the sides of the central recirculation zones and observe that the radial expansion of the jet increases for reactive conditions. The RMS uctuations feature high values in the shear layers of the ow, as reported also in Fig. 2.5 showing the LDV measurements.

Flame imaging

An intensied CCD camera (Princeton Instruments, PI-MAX 4, 1024×1024 pixels), mounted with an UV objective (Nikkor 105 mm f/4.5) is used to analyze the ame structures under steady and forced conditions. A narrow-bandpass lter (Asahi Spectra Co., ZBPA310) centered at 310 nm and with a 10 nm bandwidth is placed in front of the camera to detect the OH* chemiluminescence. Figure 2.10 rst shows a grid which is used in the image post-processing to detect the main dimensions of the ame. Figure 2.11 then shows examples of the OH* chemiluminescence signal recorded with the ICCD camera for a geometrical conguration with the swirler SW 3 , D o = 20 mm, C = 10 mm, a bulk ow velocity of U b = 5.44 m/s and an equivalence ratio φ = 0.82. Results are shown for steady (top) and forced (bottom) conditions. In this second case, the forcing frequency is f = 185 Hz, the forcing level is u /ū = 0.72 RMS and only a selected phase in the forcing cycle is shown. For the phase-locked images, the ICCD camera is synchronized with the signal driving the loudspeaker (Fig. 2.1). A long exposure time of 20 ms is used to record the mean images. The integration time needs to be reduced for phase-conditioned measurements in pulsed conditions with respect to the acoustic forcing signal. In these cases, an exposure time not exceeding T /120 was always selected, where T = 1/f is the period of the harmonic forcing signal at the frequency f . In both cases, a hundred snapshot are averaged to obtain the nal result.

The chemiluminescence signal shown in the left of Fig. 2.11 is integrated in the line of sight. Assuming that the ame is axisymmetric in average, an Abel deconvolution can be applied to the chemiluminescence images to get an information on the OH * light distribution in an axial slice of the ame in the symmetry axis [START_REF] Poularikas | Transforms and Applications Handbook[END_REF]. This is useful during images post-processing, for instance it allows to better determine the angle of the reaction layer at the base of the ame. An home-made Matlab routine is used to compute Abel transforms. Results are shown in Fig. 2.11-right. The high level of noise along the central axis is due to the mathematical formalism of the Abel transform that yields a singularity along the symmetry axis [START_REF] Poularikas | Transforms and Applications Handbook[END_REF]. Figure 2.12 shows 6 phase averaged pictures covering a full cycle of oscillation. The phase angle indicated on each picture in Fig. 2.12 is referred to the velocity signal recorded by the hot wire. The phase average imaging procedure is performed as follows.

When the ame is submitted to harmonic acoustic forcing, it oscillates at the forcing frequency. The harmonic forcing signal provided by the loudspeaker, is also used to trigger the ICCD camera. Each image in Fig. 2.12 is obtained by adding a constant time oset to the triggering signal. By recording simultaneously the velocity signal measured by the hot wire, each image is related to an instant in the cycle, as shown at the bottom in Fig. 2.12.

Numerical setup

This chapter is dedicated to the description of the numerical setup used throughout this work for the comparison between LES and experimental data. The code selected for the numerical simulations is briey described. Simulations are carried out for one selected geometry of the injector, which is described in the second section of the chapter; together with details on the mesh and the boundary conditions applied.

A comparison between numerical simulations and measurements for the ow in absence of acoustic forcing is presented in the last section of the chapter.

Large Eddy Simulations with AVBP

Large Eddy Simulations (LES) are dedicated to solve the turbulent compressible Navier-Stokes equations and are an intermediate numerical concept between Direct Numerical Simulations (DNS) and the Reynolds Averaged Navier-Stokes (RANS) methodology [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF].

A DNS solves the full instantaneous Navier-Stokes equations explicitly, without any modeling. RANS simulations compute only averaged values, without solving turbulent quantities. LES is an intermediate concept, since it resolves the large turbulent scales and models the small scales by applying a lter [START_REF] Poinsot | Theoretical and Numerical Combustion[END_REF]. RANS, LES and DNS properties are summarized in terms of energy spectrum in Fig. 3.1. All spatial frequencies in the spectrum are resolved in direct numerical simulations. Only the largest ones, up to a cut-o wave number k c , are computed in LES while the others are modeled. Turbulence is fully modeled in RANS.

In this thesis, LES are carried out with AVBP, a parallel CFD code developed by Cerfacs in partnership with IFP Energies Nouvelles [START_REF] Schönfeld | Steady and Unsteady Flow Simulations Using the Hybrid Flow Solver AVBP[END_REF][START_REF] Moureau | Numerical methods for unsteady compressible multicomponent reacting ows on xed and moving grids[END_REF][START_REF] Sengissen | LES and experimental studies of cold and reacting ow in a swirled partially premixed burner with and without fuel modulation[END_REF]. AVBP solves the three-dimensional laminar and turbulent compressible non-reactive and re- active Navier-Stokes equations, handling unstructured grids of any cell type. The code is not described here, as no developments have been made in this code during this work. The interested reader is referred to the cited references. Calculations are performed only for a non-reactive case and the AVBP code was chosen for its ability to retrieve the largest unsteady ow structures and the propagation of acoustic waves, two mandatory characteristics for the study that we wish to conduct.

Geometry, mesh and boundary conditions

The geometry sketched in Fig. 3.2 is selected for comparison between numerical simulations and measurements. The main geometrical parameters of the conguration are the following. The swirler vane SW 3 is selected. It is set at δ = 50 mm upstream the injector outlet section. The injector outlet diameter is D o = 20 mm. The injector unit is also equipped with the 6 mm diameter central blu body with a C = 10 mm diameter end piece. One is referred to Fig. 1.3 for a detailed view of the injector geometry.

The inlet and outlet of the simulated domain correspond, respectively, to the end of the rst convergent unit and the exit of the convergent exhaust on top of the combustion chamber, as shown in Fig. 3.2. Given that the rst measurement point for comparison with experiments is at the hot-wire location, it seems reasonable to not take into account the long upstream plenum in the simulation and thus reduce the computational cost. The region of the experimental setup, corresponding to the simulated geometry, together with the uid A cut of the simulated geometry showing the mesh is also displayed in Fig. 3.3.

The same gure also shows a zoom, focusing on the rened zones in the injection tube and at the chamber inlet. Full credits for the mesh of this geometry are to be given to F. Dupuy, a Ph.D. student at Cerfacs collaborating on this project by making reactive simulation of this same conguration. The fully unstructured grid is composed of 19.1 million cells corresponding to 3.4 million nodes, with renement regions inside the swirler vanes and in the injector. The six radial vanes of 6 mm diameter are resolved by circa 1 million elements.

The full injector, from the swirler up to the combustion chamber backplane, is resolved by about 5.5 million cells.

A small dierence between the geometrical conguration depicted in Fig. 1.3 and that in Fig. 3.2 is the distance between the top cone of the central blu body and the chamber backplane. This distance is equal to 2.5 mm in Fig. 1.3a, since most of the injector congurations analyzed in this work feature a central insert with the tip of the top cone lying 2.5 mm above the combustion chamber backplane. The geometry simulated with AVBP features a slightly shorter distance of 1.5 mm, due to a reduced length of the central rod. All the remaining dimensions of the burner are not modied. This small geometrical dierence is due to the fact that the NoiseDyn setup was shared between this thesis work and that of R. Gaudron [START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF]. In Gaudron's work, this slightly shorter central rod is used. Many measurements conducted in common with R. Gaudron at the beginning of this work were done on this setup. This setup has also already been used as a validation tool for the numerical simulations conducted in the thesis work of M. Merk (Merk, 2018). Due to the abundance of experimental data on the ow eld with a rod tip 1.5 mm above the injector plane, we also selected this geometry for the simulations conducted in this thesis work and in that of F. Dupuy working at CERFACS. In this work, simulations are conducted only for cold ow conditions, while F. Dupuy is working on simulations in reactive conditions. In the following, only when there could be a doubt about the value of the distance between the cone and the chamber backplane, its value will be specied.

The inlet and outlet boundary conditions are imposed through the non-reecting Navier-Stokes Characteristic Boundary Condition (NSCBC) formulation [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous ows[END_REF], which avoids acoustic reection by the use of a relaxation coecient [START_REF] Selle | Actual Impedance of Nonreecting Boundary Conditions: Implications for Computation of Resonators[END_REF]. This is an important aspect for the acoustically pulsed simulations presented in Chapter 7. The relaxation coecient should be higher than zero in order to avoid drift of mean quantities [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous ows[END_REF], but low enough to avoid acoustic reections on the boundary. Values used for reection coecients are indicated in Tab.

Q = 2.07 m 3 s -1 (RELAX_UVW_T_Y) T = 293 K Outlet Target pressure 100s -1 p = 101325 Pa (RELAX_P_3D)

Walls

No-slip adiabatic mean and oscillating quantities. At the inlet of the domain, a top-hat velocity prole like the one found in the experiments, is imposed. As in the experiment, the bulk ow velocity at a pressure p atm = 101325 Pa and at a temperature T = 293 K, is set to a value U b = 5.44 m/s. At the outlet of the domain the same atmospheric pressure is prescribed. Finally, all the solid walls are treated with a no-slip adiabatic boundary condition (Tab. 3.1). No attempt was made in this work to impose the correct thermal boundary condition [START_REF] Mercier | Experimental and numerical investigation of the inuence of thermal boundary conditions on premixed swirling ame stabilization[END_REF] as only cold ow simulations are envisaged.

The numerical integration relies on a two-step Taylor-Galerkin weighted residual central distribution scheme with a third order precision in space and time [START_REF] Rudgyard | Development of High-Order Taylor-Galerkin Schemes for LES[END_REF]. The Sigma model, developed by Nicoud et al., 2011, is used as a subgrid scale model for turbulent uctuations.

Comparison with experiments

Simulation conducted with this numerical setup are validated against experimental data in the absence of external forcing. For these comparisons, the averaging LES time amounts to 50 ms. At rst, LES data are compared to LDV measurements in Fig. 3.4. It is important to remember that the LDV measurements are conducted without connement for optical reasons. Nonetheless, the average axial, azimuthal and radial velocity proles obtained with LDV or LES compare well in the left of Fig. 3.4. The LES well captures both the shape and the amplitudes of the velocity proles. The root-mean-square (rms) values are also compared in the right of Fig. 3.4. These quantities are generally more dicult to capture by LES. The shape of the rms proles are here well captured by the simulation. The rms velocity values show a peak in the shear layers, close to the rim of the injector and near the central cone, in both the LDV and LES data. The rms velocities are underestimated by LES, especially in the central region of the jet in the wake of the blu-body. One reason might be that the relatively short time in the LES over which averages were made is not long enough to converge toward the correct average values in this region characterized by very low recirculation velocities. The overall comparison yields, however, satisfying agreement for the cold ow condition explored.

LES data are compared to PIV measurements in Figs. 3.5 to 3.7. Figure 3.5 shows velocity elds in an axial plane (top) and in a transverse plane at a distance of 2 mm from the blu-body. To take into account the fact that the PIV laser sheet has a thickness of about 1.5 mm, the LES results are obtained by averaging eleven equally separated cuts, from -0.75 mm to +0.75 mm with respect to the central one. The transverse scans in Fig. 3.5, show the velocities in the x direction. For a coordinate x = 0, this velocity would correspond to the azimuthal velocity, while for y = 0 it corresponds to the radial velocity. Azimuthal and radial velocity proles extracted from PIV and LES are compared in Fig. 3.6. These pictures show these proles at a distance of 2 mm downstream the blu-body. The shape of both proles match very well, and only a slight overestimation of the peak velocities is observed with LES results. The axial velocity scans match well in the top of Fig. 3.5. One needs to pay attention to the fact that, in these two pictures, the coordinate z starts at a distance of 2 mm from the blu-body, corresponding to the rst point measured by PIV. The inner recirculation zone with negative velocities is well captured by the LES. Also the spreading of the jet is well retrieved. This is conrmed in Fig. 3.7, showing radial proles of the axial velocity at dierent distances z = 5, 10, 15, 20, 25 and 30 mm above the tip of the central blu body. Axial velocity proles obtained by PIV or by LES compare well.

Conclusions

LES are performed with the AVBP code and compared with measurements for one conguration of the injector featuring swirler SW 3 , an injector outlet diameter D o = 20 mm and a central insert ended by a cone of C = 10 mm. In the explored conguration, the conical end piece protrudes 1.5mm in the combustion chamber. The same conguration was used in the thesis of R. Gaudron and M. Merk [START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF]Merk, 2018). LES well reproduce the cold ow eld of the conguration explored in the axial and transverse planes at which measurements and simulations were compared. This numerical setup is also used in Chapter 7 to investigate how a modulation of the ow rate alters the injector dynamics and the ow eld in the combustion chamber.

Chapter 4

Flow and ame propertieswithout acoustic coupling

This chapter is dedicated to the analysis of steady mean ow and ame properties. A detailed investigation of the eects of modications of the swirler and injector geometries is conducted. At rst, a stabilization chart is obtained, separating three main regions where ames are respectively ash-backing into the injector, well-stabilized in the combustion chamber or have an elongated shape. These dierent shapes depend on a normalized ow passage area and on the swirl number.

The analysis for well-stabilized ames is then taken to the next level, by analyzing the mean axial and transverse velocity elds and the ame structures, for some selected congurations.

Burner stabilization chart

The experimental setup used for the analyses described in this chapter is shown in Fig. 1.3. It was fully described in Chapter 1. Each injector could be equipped with a dierent swirling vane SW placed at a distance δ from the injector outlet, a dierent nozzle tube of diameter D o and a dierent conical end piece of diameter C at the top of the central blu body.

The dierent congurations analyzed are summarized in Table 4.1. Five dierent radial swirlers with tangential inlets, described in Section 1.2.1, are tested together with dierent values of the injector exit diameter D o = 18, 20 and 22 mm and of the conical end piece diameter C = 10, 12 and 14 mm. These geometrical modications lead to dierent values of the cross section area A 2 at the exit of the injector, while the cross section area A 1 380 mm 2 at the hot wire location is xed for all congurations.

It is known that the swirl number S controls the stabilization of swirling ames 70 Chapter 4 -Flow and flame properties -without acoustic coupling

S = 2 D o u θ u z r 2 dr u 2 z rdr . (4.1)
The values found for the swirl number are also indicated in the last column in Table 4.1 with a relative precision ±3% taking into account uncertainties on the measurements. For all other remaining congurations tested in this study, the analytic expression Eq. ( 1.5) is used to estimate the swirl number S * . We rst consider the case when the swirler design SW is modied, but the cross-section area at the exit of the injector A 2 is xed. The mean axial velocity u z prole does not change, while the peak of the mean azimuthal component u θ augments from SW 1 to SW 2 and shifts to a slightly higher radial position. This is due to the increase of the distance 2H between the axes of the tangential inlets of the swirling vane. This leads to an increase of the swirl number S for SW 2 compared to SW 1 . We now examine the case when the swirler design SW is xed, but the cross-section area at the exit of the injector A 2 is reduced. This corresponds to the data in red and green in Fig. 4.1. The azimuthal velocity u θ prole does not change much in this case, while there is a clear increase of the peak axial velocity u z , which leads to a strong reduction of the measured swirl number S between the two cases. From Table 4.1, one can see that the swirl number increases from swirler SW 0 to SW 4 when the cross-section area at the exit of the injector A 2 remains constant. Once again, this is due to the increase of the distance 2H between the axes of the swirler tangential inlets (see Fig. 1.4). For a xed swirler design, the swirl number S increases if the injection nozzle diameter D o increases or if the diameter C of the conical end piece decreases, i.e. when the ow passage area A 2 in the injection tube increases. This leads to a decrease in axial velocity with almost constant azimuthal velocity.

Flame stabilization is now investigated for the dierent injectors tested. Three stabilization modes are identied depending on the value of the swirl number S and the normalized injector nozzle cross section area A 1 /A 2 . These three regimes are presented in Fig. 4.2. Close to blow-o, when the swirl number is too low, the ame is elongated in the downstream direction as in the left image in Fig. 4.2 (a). In the middle image, when the swirl number is high enough, the ame is well stabilized inside the combustion chamber and has a compact "V" shape. In the right image, when the swirl level is too high, ash back takes place and the ames protrudes inside the injector.

The stabilization chart in Fig. 4.3 delineates the dierent regimes observed for the injectors tested in Tab. 4.1. When the swirl number is between 0.5 and 0.8, ames are well-stabilized over a wide range of injector dimensions and for a wide set of swirler designs. When S < 0.5, the shape of the ame becomes elongated and one approaches the blow-o limit. When S > 0.8, ashback inside the injector can occur. As shown in Fig. 4.3, these critical boundaries also depend on the ratio A 1 /A 2 . It is worth recalling that these experiments were conducted at a xed equivalence ratio φ = 0.82 and bulk ow velocity U b = 5.44 m/s and these boundaries obviously depend on these operating conditions and on the setup used in this study.

Mean ow properties

The mean ow properties are analyzed by comparing PIV results in an axial plane and in a transverse plane for four geometrical congurations of the injector as shown in Fig. 4.4. The rst two rows in Fig. 4.4 show results obtained for xed injector geometries with D o = 22 mm and C = 14 mm but dierent swirler designs SW 0 and SW 3 and leading to dierent swirl numbers S = 0.20 and S = 0.80. Swirler SW 0 features radial injection channels aligned with the radial direction and, as a consequence, velocities in the transverse plane are much lower compared to those measured for SW 3 , featuring injection channels shifted from the radial direction. This is also highlighted in Fig. 4.6-left, showing the azimuthal velocity proles obtained from the transverse PIV results at 2 ± 0.5 mm above the blu body. The central recirculation region is concentrated in a small area in the wake of the central blu-body in the top row of Fig. 4.4, while it is much larger and extends further downstream in the second row obtained for swirler SW 3 . When the swirl number is high enough, 4.4-4.5. This is due to small imperfections in machining and diculty to center the blu body over a relatively long distance from the swirler backplane, where it is screwed, to the injector outlet. Results for SW 0 are presented in blue in Fig. 4.5. Close to the chamber backplane, the IRZ strength is comparable for SW 0 and SW 3 (shown in red). But it is clear that the IRZ of SW 0 becomes weaker as the distance from the chamber backplane increases. At a distance of 15 mm, the axial velocity is positive all along the scan for SW 0 in Fig. 4.5, while negative velocities in the central region of the jet are observed for all the locations considered for SW 3 . The angle of the jet is bigger for the conguration with SW 3 than with SW 0 . This is also seen in Fig. 4.6-right, showing the radial velocity proles obtained from the transverse PIV scan at 2 ± 0.5 mm above the blu body. The radial velocity is much higher for SW 3 than for SW 0 in this gure.

Images in the third and fourth rows in Fig. 4.4 show the velocity eld for congurations featuring the same swirler SW 3 , but dierent values of the cross section area A 2 at the injector exit. These modications were obtained by substituting the central injection nozzle D o = 18 -20 mm and the blu body end piece C = 12 -10 mm. The swirl number is S = 0.80 for the conguration shown in the third line, while it is reduced to S = 0.60 for that in the fourth line. From Fig. 4.6-left, one can check that this is not due to a change of the azimuthal ow component. As already seen in Fig. 4.1, when the swirler design is xed, a change of other geometrical parameters does not inuence much the azimuthal velocity ūθ proles. The swirl number decrease is here due to a dierent shape of axial velocity proles, plotted in orange and green in Fig. 4.5. When the size of the injector and blu-body are modied, but the normalized cross section A 1 /A 2 remains almost unchanged, as for the results shown in the second and third row in Fig. 4.4, the axial velocity proles are almost superposed in Fig. 4.5 (red and orange curves). When the cross section at the exit of the injector is reduced, the axial velocity prole shifts to higher values (green curve in Fig. 4

Mean ame structure

Eects of the swirl number S on the shape taken by the ames are shown in Figs. 4.7 and 4.8. It is rst worth recalling that the equivalence ratio φ = 0.82 is xed for all these experiments. These images are obtained by averaging a hundred snapshots, each with a long exposure time of 20 ms, that are recorded with the ICCD camera equipped with the OH* lter. Here again, a small asymmetry can be identied in the light distributions between the left and right part in these images.

Some general features about the shape of the ames can be inferred from these gures, obtained for several dierent injection conditions. As the swirl number decreases, either by a reduction of the injector exit diameter D o (rst row in Fig. 4.7) or by a increase of the blu-body end piece diameter C (second row in Fig. 4.7), the ame becomes more elongated, moves further away from the injector outlet and closer to the chamber walls. This is due to the increase of axial velocity, as shown in Figs. 4.1-4.5. In the third line of Fig. 4.7, eects of the swirler design are highlighted. As the swirl level S decreases from 0.95 to 0.65, the ame length and its leading edge position with respect to the burner outlet remain roughly unaltered, while the ame angle with respect to the axial direction reduces. This is due to the strong change in azimuthal velocity imposed by the dierent swirler SW designs (Figs. 4.1-4.6), when all the other geometrical parameters are kept xed. When the swirl number is reduced to S = 0.20 for SW 0 , the light distribution takes a relatively long and narrow shape concentrated in the wake of the central blu-body.

The OH* light distribution does not feature any visible change when the swirler is mounted at dierent distances δ with respect to the combustion chamber

Conclusions

The experiments reported in this chapter, done in the absence of acoustic forcing, show how to control the structure of the resulting ow eld and ame shape when the injector design is modied by either changing the nozzle diameter, the top cone end piece diameter or the swirling vane. It was shown that three dierent regimes could be obtained, with elongated ames close to blow-o, compact well stabilized ames and ames ash-backing inside the injector. The swirl level has been shown to largely determine the structure of the ow eld and the resulting ame shape. The way the swirl level determines these properties, is by modifying the axial, azimuthal and radial velocity proles in the combustion chamber.

A last set of experiment was conducted with modication of the swirling vane distance δ with respect to the injector outlet or of the bulk ow velocity U b . It was shown that a change of the distance δ does not impact the ame structure, while an increase of the bulk ow velocity U b increases the ame size but keeps the same ame aspect ratio. This is due to the fact that these two quantities are not expected to modify the swirl number S, once again conrming that the structure of the ow eld and the ame shape largely depends on this quantity.

Chapter 5

Characterization of thermoacoustic instabilities

The test rig equipped with dierent exhaust tubes and injection devices features a set of self-sustained thermoacoustic instabilities. This chapter is dedicated to their analysis. After a brief description of the specic diagnostics used for the analysis of these instabilities, the rst part of this chapter is dedicated to the characterization of one unstable regime.

The rest of this chapter is dedicated to the description of the eects of modications of the injector and swirler designs on the stability of the combustor.

Analysis of a naturally unstable regime

Lean premixed combustion systems achieve low pollutant emission levels, with compact ames and high power densities, but are sensitive to external perturbations. The absence of perforated liners, which are present in non-premixed designs, reduce the damping of premixed systems. These reasons make thermoacoustic instabilities one of the major drawbacks, that hinder the practical operation of lean premixed combustion systems. This matter has been studied since the early fties [START_REF] Crocco | Aspect of Combustion Stability in Liquid Propellant Rocket Motors Part I[END_REF][START_REF] Crocco | Aspect of Combustion Stability in Liquid Propellant Rocket Motors Part II[END_REF] but, nonetheless, progress is still needed if one want to be able to predict and eventually control the onset of instabilities. In a simplied conguration, like the one investigated in this work, where the mixture composition remain uniform, combustion instabilities result from a resonant coupling between ow rate oscillations, unsteady combustion and acoustics [START_REF] Ducruix | Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms[END_REF][START_REF] Candel | Dynamics of Swirling Flames[END_REF].

The test-rig used for the analysis of thermoacoustic instabilities is sketched in Fig. 5.1. Only the components and instruments specic to this analysis are highlighted in this gure. A detailed description of the setup was already given in chapter 1. A mixture of air and methane is injected from two diametrically opposed apertures at the base of the burner. The air and methane mass ow rates where normally the hot wire probe is placed. Microphones MC and ME in Fig. 5.1 are used to measure the pressure at the combustion chamber backplate and in the exhaust tube. In this hot gases region, microphones are mounted on water-cooled waveguides. More information on these waveguides are given in the Ph.D. thesis of [START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF] All microphones are connected to a conditioning amplier (Bruel & Kjaer, 2690).

The length of the exhaust tube downstream the second convergent unit can be modied by mounting up to three tubes of length E = 220 mm (Fig. 5.1). The conguration without any exhaust tube (E = 0 mm in Fig. 5.1) is the one used in chapter 4 to analyze the mean ame properties in the absence of acoustic forcing and is used here to get an estimate of the thermalization time of the system.

Three K-type thermocouples are used for temperature measurements. TC 1 is inserted in the combustion chamber through the pressure tap for microphone MC. It is used to estimate the burnt gases temperature, at a distance of approximately 2 ± 0.5 mm from the chamber backplane. TC 2 is located at approximately half the height of a quartz side window and inserted in one of the lateral steel rod at the corner of the combustion chamber. TC 3 is put in contact with the second convergent unit, at approximately half of its height.

It is important to underline that these measurements are not intended to give a precise indication of the temperature at these locations, but are just used to get an indication of the system thermalization time.

Results of temperature measurements are shown in Fig. 5.2, with respect to the ignition time of combustion. After a rapid initial increase in the rst 10 minutes, the temperature of combustion reaches a steady-state value at about 20-25 minutes. All the measurements conducted in this work in reactive con- ditions, for instance the ICCD imaging to analyze the ame structure in the absence of forcing as in chapter 4 or the FTF measurements presented in chapter 6, were done after 25 minutes from the ame ignition, to let the system reach its thermalization.

When the length of the exhaust tube downstream the second convergent is changed, the thermoacoustic state of the burner is modied. Without any exhaust tube (E = 0 mm in Fig. 5.1), the system is stable, and no distinct tone emerges from the pressure spectrum. This was conrmed for all the dierent geometrical congurations tested by modifying the injection unit. The conguration with an exhaust tube length of 2E = 440 mm is now analyzed in more detail. The evolution of the SPL measured by microphone MC is plotted in Fig. 5.6-left at dierent times after ignition. The SPL rapidly builds up in the rst minutes of the experiments to reach a maximum of 148 dB inside the combustion chamber around 7-8 minutes after the ame ignition in Fig. 5.6-left. In this rst part of the experiment, the combustion chamber strongly vibrates leading to a large mechanical strain on the sealing between the quartz windows and the metal rods. It was checked that there was no air leakage from the chamber during the experiments and all measurements presented were conducted at least two times to check their repeatability. It was necessary to rebuild periodically the combustion chamber after each characterization to ensure a perfect joint sealing for subsequent experiments.

After the initial increase, the maximum SPL measured by MC drops a little to reach a plateau of about 140 dB for t ≥ 20 minutes when the system has reached thermalization. The frequency f I of the instability plotted in Fig. 5.6right also increases with time. It reaches a limit cycle at f I 185 Hz after t ≥ 20 minutes. This analysis conrms the strong impact of thermal boundary conditions on the solid components of the combustion chamber [START_REF] Mercier | Experimental and numerical investigation of the inuence of thermal boundary conditions on premixed swirling ame stabilization[END_REF] on its thermo-acoustic state and in particular on the frequency and level reached by the instability at limit cycle. The pressure spectra measured by the dierent microphones presented in Fig. 5.7 clearly feature a main peak at the instability frequency, but the level of the pressure uctuations varies with time. Figure 5.7 also shows that the spectral distribution is altered with the second harmonic and moves as time increases highlighting how acoustic energy is redistributed.

Eect of the swirler to injector outlet distance δ

Eects of modication of the injector design on the thermoacoustic instability properties is now investigated by rst modifying the distance δ between the swirler outlet and the nozzle outlet as in Fig. 5.9. The thermoacoustic is here also modied by changing the exhaust tube length (1E, 2E, 3E). All the other parameters remain xed. It is rst recalled that the mean ame shape without any exhaust tube (E = 0 mm) is the same in these two cases as shown in Fig. 5.10. By modifying the length of the exhaust tube, the thermoacoustic state of the burner can be modied. The objective here is to analyze eects of modications of the injector design on this thermoacoustic state. Figure 5.11 shows the SPL spectrum measured with the microphone MC for δ = 35 mm (Fig. 5.11-left) and δ = 50 mm (Fig. 5.11-right) with three dierent lengths of the exhaust tube (1E, 2E, 3E). When the swirler is mounted at a distance δ = 35 mm from the chamber, a weak instability with a maximum SPL of 127 dB is observed at f I = 220 Hz for the conguration with an exhaust tube of length 1E = 220 mm. By increasing this length to 2E = 440 mm, the maximum SPL is reduced to 124 dB and the frequency of the instability drops to f I = 196 Hz. When three tubes are mounted downstream the combustion chamber, the system is stable for 3E = 660 mm. The situation is reversed in Fig. 5.11-right when the swirler is mounted at δ = 50 mm from the nozzle exhaust section. A much stronger instability is observed for the conguration with an exhaust tube of length 3E = 660 mm with a maximum SPL of 142 dB at f I = 164 Hz. The SPL slightly drops to 140 dB at f I = 185 Hz for a 2E = 440 mm exhaust length, but the system is still strongly unstable in this conguration, as described in section 5.1. The system is now stable for an exhaust length of 1E = 220 mm. In Fig. 5.12, the pressure uctuations in the combustion chamber measured by MC are plotted along with the heat release rate uctuations measured by the photomultiplier PM in Fig. 5.1 equipped with an OH* lter. Results are shown for the conguration with an exhaust length of 2E = 660 mm for the two distances δ = 35 mm and 50 mm. Both signals in left and right plots in Fig. 5.12 oscillate at the same frequency. The oscillations for the combustion pressure and heat release rate are much larger for the conguration with δ = 50 mm shown on the right plot in Fig. 5.12 than when the swirler is set at δ = 35 mm. This was already seen in Fig. 5.11, showing that the instability is stronger when two tubes are mounted downstream the chamber and the distance δ is increased from 35 mm to 50 mm. One can conclude that this is not due to a dierent phasing between pressure and heat release rate uctuations. One clearly sees in Fig. 5.12 that these signals are oscillating in phase in the left plot for δ = 35 mm and right plot for δ = 50 mm. The drop of pressure oscillations is thus not due to a shift of the heat release rate signal with respect to the pressure signal as the distance δ is reduced.

These observations conrms that small changes of the injector design can lead to dramatic changes of the thermoacoustic state of the system. An interpretation of the mechanisms leading to this dierences in strength of the instability for dierent values of δ is given in chapter 6 with the help of FTF measurements.

Eect of the swirling vane

The investigation now proceeds by changing the swirling vane design SW . Measurements are conducted with the longest exhaust tube, 3E = 660 mm. Two dierent radial swirlers SW 1 and SW 3 are tested. Their design is described in chapter 1. All the remaining geometrical parameters of the injector are xed: D o = 20 mm, C = 10 mm, δ = 50 mm. The bulk ow velocity U b = 5.44 m/s and equivalence ratio φ = 0.82 remain also unchanged. The swirl level with this design and the radial vane SW 3 is S = 0.80 and it drop to S = 0.69 when SW 3 is replaced with SW 1 .

The sound pressure spectrum measured by MC in the combustion chamber at a time t = 40 minutes after ignition is presented in Fig. 5.13-left. The pressure spectrum is basically superposed for the two swirler designs. The evolution of the instability frequency f I and of the maximum SPL measured by MC is presented in Fig. 5.13-right. Data are presented only after the system has reached its thermalization. The frequency of the instability is close to f I 164 Hz for both the explored congurations. The maximum SPL measured by microphone MC is close to 142 dB and is close for the two swirler designs while the ame shape in the absence of acoustic forcing diers, as shown in Fig. 5.14 for a slightly dierent injector geometry.

These results show that the swirler design does not play an important role itself on the properties of the thermoacoustic instability, when all the other parameters are kept xed. An interpretation of these observations is given in chapter 6 with the help of FTF measurements.

Conclusions

The thermoacoustic state of the test rig developed in this work has been analyzed when the combustion exhaust length is modied and some components of the injector are replaced. Experiments have been conducted at xed equivalence ratio φ = 0.82 and bulk ow velocity U b = 5.44 m/s.

It has rst been shown that the thermal state of the combustor strongly alters the thermoacoustic state of the system. This has been checked for one conguration of the injector. It is found that the sound power level (SPL) inside the combustion chamber strongly builds up in the rst minutes after ignition and then reaches a plateau at about t 20 min. The instability frequency constantly increases in the rst minutes and then reaches a limit cycle for t ≥ 20 min. It is then necessary to wait for the system to reach its thermalization before one can conclude about its thermoacoustic state.

It has been shown that increasing the combustor exhaust length can lead to increasingly higher oscillation levels or lower oscillation levels depending on the distance between the swirler and injector outlet, if all other parameters are kept xed. This has been shown for two dierent values of the distance δ between the swirler exit and the combustion chamber backplane. For a swirler mounted at a distance δ = 35 mm, we observed a reduction of the SPL inside the combustion chamber for increasing length of the combustor exhaust, while the opposite was observed for a distance δ = 50 mm. It has also been shown that modifying the swirling vane does not necessarily help to control the oscillation level in the combustion chamber. This was checked for two dierent designs of the radial swirling vane. Despite the fact that the shape of the ame stabilized with these dierent swirling vanes in the absence of acoustic forcing diers, the SPL measured in the combustion chamber for the two congurations is close. These two observations are discussed in more detail in chapter 6 with the help of FTF measurements.

Chapter 6

Eects of injector design and injection conditions on Flame Transfer Functions

The eects of dierent injector geometries and injection conditions on the ame transfer function are examined in this chapter. The objective here is to describe the shape taken by the FTF and assess the main parameters altering the ame response. The next chapter is dedicated to the analysis of the mechanisms controlling this response. The chapter is organized in the following way. After a brief introduction, a typical ame describing function determined for a given injector conguration is illustrated. It is shown that the shape of this response features the same characteristics for all forcing levels. In the second part of the chapter, results are reported at a xed forcing level but for dierent geometrical congurations. The eects of modications of the distance between the swirler exit and the combustion chamber backplane are considered rst, followed by the eects of a change of the bulk ow velocity.

In the last part of the chapter, results obtained while changing the injector exit diameter, the blu-body end piece diameter or the swirler design are considered.

Introduction

In fully premixed systems and in absence of mixture composition disturbances, the ame transfer function (FTF) between heat release rate uctuations Q and harmonic velocity disturbances u that produce them is dened as [START_REF] Ducruix | Combustion Dynamics and Instabilities: Elementary Coupling and Driving Mechanisms[END_REF]:

F(f ) = Q / Q u /u = G(f ) exp(iϕ(f )) (6.1)
96 Chapter 6 -Effects of injector design and injection conditions on Flame Transfer Functions where G denotes the gain and ϕ stands for the phase lag of the FTF, both depending on the forcing frequency f . In recent developments, this linear concept has been extended to the Flame Describing Function (FDF) when eects of the perturbation level |u |/ū are explicitly considered (Noiray et al., 2008;[START_REF] Hosseini | The non-linear thermoacoustic response of a small swirl burner[END_REF][START_REF] Palies | Experimental study on the eect of swirler geometry and swirl number on ame describing functions[END_REF][START_REF] Osi¢ | Response of a swirl-stabilized ame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF]:

F(f, |u |/u) = Q / Q u /u = G(f, |u |/u) exp(iϕ(f, |u |/u)). (6.
2)

The objective of this chapter is to characterize the dierences between the FTF of swirled ames stabilized on various injection system designs. Only fully premixed ames at xed equivalence ratio φ = 0.82 stabilized on radial swirling injectors are considered. Their FTF is determined and the main elements leading to the largest drop of the FTF gain are analyzed with a set of experiments.

The design modications comprise changes of ( 1) the distance between the swirler exit and the combustion chamber backplate, ( 2 The experimental setup used to determine the FTF/FDF is shown in Fig. 6.1 and was described in detail in Chapter 1. The FTFs are determined from Eq. ( 6.1) by submitting the ame to harmonic modulations of the owrate.

The velocity uctuation u is controlled by the hot wire anemometer HW. This probe is located in the nozzle of the convergent unit in Fig. 6.1 at a distance of 40 mm upstream the swirling vane. It has been checked that the velocity has a top hat prole and is laminar at this location. The photomultiplier equipped with an OH* lter is used to determine the mean Ī and uctuating I luminosity signals integrated over the ame volume and over the line of sight. These signals are assumed to be a good tracer of the heat release rate for the lean premixed ames investigated here [START_REF] Hurle | Sound emission from open turbulent premixed ames[END_REF]. The transfer function is then deduced from the cross and power-spectral densities between the photomultiplier and hot wire signals examined at the forcing frequency f . These signals are recorded at a sampling rate of f s = 8192 Hz over 4 seconds and Welch periodograms are used to obtain statistically converged results. More details were given in chapter 2.

Eect of the forcing level -Flame Describing Function

The case considered here is also analyzed by LES and was presented in chapter 3. This geometry features the radial swirling vane SW 3 , an injector outlet diameter D o = 20 mm, a blu body with a top cone of diameter C = 10 mm and a 1 mm shorter central rod with respect to that shown in Fig. 1.3-a. The distance between the top of the cone and the chamber backplane is 1.5 mm. The bulk ow velocity is U b = 5.44 m/s and the equivalence ratio is φ = 0.82.

The mean shape of the ame stabilized with this injector conguration without acoustic forcing is shown in Fig. 6.2.

Figure 6.3 shows FDF data in the frequency range 20-250 Hz, with a 10 Hz resolution, for velocity perturbation levels ranging from u /u = 0.10 to u /u = 0.72, where u and u denote the mean and root-mean-square values of the signal measured by the hot wire HW. The full frequency range could not be covered for all the forcing levels because of power limitations of the actuation system.

The shape of the FDF in Fig. 6.3 is very similar to that found by Palies et al., 2010a and in many other studies [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF][START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. At zero frequency the FDF gain approaches unity. The FDF gain then rst increases with the frequency to reach a maximum at a frequency close to 50 Hz for all forcing levels. But the maximum gain value at this peak frequency depends on the forcing level. In particular it regularly drops as u /u increases. This is a clear manifestation of a non-linear behavior. The gain curve then features a sudden drop with a valley and reaches a minimum value at a frequency close to 120 Hz for all forcing levels. This frequency also corresponds to an inection point in the FTF phase plot. The minimum value reached by the gain does not depend on u /u. So, at the minimum gain frequency, the behavior may be considered roughly linear with the forcing level. At higher frequencies, the FTF gain increases again to reach a second maximum, around 190 Hz. Once again, this frequency is independent of the forcing level u /u and the FDF gain at this second peak frequency drops regularly as u /u increases. At zero frequency the FDF phase-lag approaches zero. The phase-lag curve then increases with an almost constant slope between 20 -100 Hz. It then features a sudden drop with an inection point around 100 -150 Hz, in the region where the FDF gain features its minimum value. At higher frequencies, the phase-lag curve still increases with a constant slope. It is important to notice that the curves obtained at all forcing levels are superposed, meaning that the delay between incoming velocity perturbations and resulting heat release rate uctuations is not a function of the forcing level u /u.

As indicated in the introduction, the objective of this chapter is to identify the main elements of the injector leading to the largest drop of the FTF gain. Figure 6.3 clearly shows that the forcing level does not lead to a change of the minimum gain value and that results are independent of the forcing level at this peculiar frequency. For this reasons, results presented in the following sections are obtained at a xed forcing level.

Eect of the distance δ and of the bulk ow velocity U b

Eects of modications of the distance δ between the swirler exit and the combustion chamber backplate on the ame response are now analyzed. This distance is modied here between δ = 35-50 mm, while it is xed to 50 mm for 100 Chapter 6 -Effects of injector design and injection conditions on ical end piece diameter C = 12 mm. The shape of these ames in the absence of forcing is reported in Fig. 6.5, showing that the ame size increases, keeping the same aspect ratio, as the injection velocity U b is increased. The swirl number S = 0.73 was measured only for the conguration with U b = 5.44 m/s and is here assumed to remain the same for the other congurations with a lower velocity. FTF results for these ames are reported in Fig. 6.6-right at a forcing level u /u = 0.30 RMS.

The FTFs reported in the left and right plot in Fig. 6.6 are both characterized by alternating regions with rst a maximum and then a minimum value of the gain as the excitation frequency increases. Their phase lag also features an inection point around the frequency of the gain minimum. One retrieves the same features observed for the FDF shown in Fig. 6.3. The peculiarity of the results shown in Fig. 6.6 is that the FTF gain and phase-lag curves are basically shifted in the frequency range, without any important modication of their shape, as δ or U b are modied. The minimum value reached by the FTF gain remains the same in Fig. 6.6-left for the dierent cases explored with changes of the distance δ and in Fig. 6.6-right for the dierent cases explored with changes of the bulk ow velocity U b , but the frequency corresponding to this minimum gain value changes as δ or U b are modied. In particular, this frequency increases as δ is reduced or as U b is increased. This is due to a change of the time required for the convected disturbances generated at the swirler trailing edge to reach the burner outlet. This time lag decreases as the distance δ decreases or as the bulk velocity in the injection tube increases. It is again interesting to observe that the ratio of f and U b is remains also constant in this three cases. This is highlighted in Fig. 6.7-right, showing the evolution of the frequency corresponding to a minimum FTF gain value with respect to the bulk ow velocity U b .

6.4 Eect of geometrical modications of the injector unit In these cases the swirl levels are S = 0.69 and S = 0.63 respectively. The drop of the FTF gain is the largest for the biggest injector diameter D o = 22 mm. In this case, a gain as low as G 0.15 is measured for SW 2 (S = 0.79), while the minimum gain is G 0.5 for swirler SW 1 (S = 0.71). FTF measurements for swirler SW 3 were attempted in the same conguration. Results cannot be presented since, for this geometry featuring a swirl number S = 0.81, the ame ashed back into the injection tube when submitted to owrate modulations at frequencies close to 100 Hz even for a forcing level as low as u /u = 0.05 RMS.

At higher forcing frequencies the FTF gain increases again in Fig. 6.8 to reach a second maximum. It is worth noting that the maximum gain values of the FTF obtained with swirlers SW 1 and SW 2 are close in Fig. 6.8 when the diameter D o is the same. The dierences in the amplitude response between these two swirler designs are the largest at the frequency corresponding to the minimum FTF gain. It is also interesting to observe that the frequencies corresponding to this peculiar minimum values of the FTF gain are slightly shifted to higher values as the injector nozzle diameter D o is reduced and the ow velocity in the tube between the swirler exit and the chamber backplane increases. This is once again due to a change of the time required for the convected disturbances generated at the swirler trailing edge to reach the burner outlet. This time lag decreases since the bulk velocity in the injection tube increases.

The phase-lag curves in Fig. 6.8 also feature a sharp transition with an inection point around the FTF gain minimum frequency. This transition is sharper when the FTF gain drop is higher. In the rest of the frequency range investigated, the phase-lag increases regularly with frequency, showing a constant slope that increases as the injector diameter D o increases.

Eect of the blu-body end piece diameter C

Figure 6.10 shows eects of modications of the conical blu-body diameter C = 10, 12, 14 mm on the FTF for ames stabilized with the radial swirler SW 2 and an injector diameter xed at D o = 20 mm. The mean shape taken by these ames is reported in Fig. 6.11. One reminds that as the swirl number decreases, here by increasing the diameter C, the ame becomes more elongated, moves further away from the injector outlet and closer to the chamber walls. 106 Chapter 6 -Effects of injector design and injection conditions on Flame Transfer Functions

It is important to underline that, for all these congurations sharing the same ow velocity in the injection tube between the swirler exit and the chamber backplane, the frequencies corresponding to minimum or maximum gain values are exactly the same.

The shape of the phase-lag curves in Fig. 6.10 is the same as in Figs . 6.3 and 6.8. The results obtained for dierent diameters C in Fig. 6.10 are roughly superposed. The only dierence is that the sharp change at the inection point where the FTF gain reaches a minimum increases as the swirl level increases and the minimum FTF gain drops.

Eect of the swirling vane

Eects of modications of the swirling vane SW 0 , SW 1 , SW 3 and SW 4 on the FTF are now presented in Fig. 6.12, for ames stabilized with the nozzle diameter D o = 22 mm and a conical end piece diameter C = 14 mm. Results are plotted in the frequency range 20 -250 Hz, with a 10 Hz resolution and for a velocity perturbation level xed to u /u = 0.30 RMS in this case. The mean shape taken by these ames is reported in Fig. 6.13.

The FTFs obtained with swirlers SW 1 , SW 3 and SW 4 have the same characteristics as those shown in Figs. 6.3, 6.6, 6.8 and 6.10. The FTF gain features an alternating behavior with rst a maximum followed by a minimum value. The minimum gain decreases as the swirl level S increases when SW 1 (S = 0.65) is replaced by SW 3 (S = 0.80). The lowest gain is obtained for the highest swirl number S = 0.95 for SW 4 .

The ame stabilized with the low swirl injector SW 0 (S = 0.20) behaves in a singular manner. In this case, the FTF gain curve G(f ) smoothly increases to reach a maximum G 1.27 at f 100 Hz and then smoothly decreases towards zero as the forcing frequency further increases. This low-pass lter behavior, with gain values exceeding unity in the frequency range 40 -150 Hz, is a well known feature of the acoustic response of laminar premixed non-swirling V-ames, as rst theoretically demonstrated in [START_REF] Schuller | A unied model for the prediction of laminar ame transfer functions: comparisons between conical and V-ame dynamics[END_REF]conrmed experimentally in Durox et al., 2005. The same type of smooth behavior with large gain overshoots at low frequencies is also a characteristic of the response of turbulent non-swirling ames [START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF]. The shape of this ame is actually closer to that of a non-swirling ame, as shown in Fig. 6.13. Indeed, the swirl number S = 0.20 is very low in this case.

The FTF phase-lag of the low swirling ame SW 0 also behaves like that of a non-swirling V-ame. It regularly increases in Fig. 6.12 with an almost constant slope in the full frequency range investigated. There is no inection point in this case. The FTF phase lag increases linearly also for the other swirling ames presented in Fig. 6.12, except in the frequency region close to the minimum value of the FTF gain, where an inection point and a rapid change of the phase lag characterize the responses of the highly swirled ames.

Flame Transfer Functions

Discussion

Some general features about the FTF/FDF of highly swirled ames (S ≥ 0.60) can be drawn from the results presented in this chapter.

The ame response is characterized by alternating regions with maximum and minimum gain values. With respect to the forcing level, Fig. 6.3, the behavior of the FTF gain is strongly non-linear around the maximum response and roughly linear at the minimum response. The FDF phase-lag does not depend on the forcing level.

When the time required for the convected disturbances generated at the swirler trailing edge to reach the burner outlet is modied, either by a change of the distance δ between the swirler exit and the combustion chamber backplane or by a change of the bulk ow velocity U b , but all the other geometrical parameters of the injector unit are xed, Fig. 6.6, the FTF gain curve is shifted in the frequency range without any important change of its shape. In particular, the minimum gain level is constant for these congurations with a constant swirl number.

When the injector geometry is modied in such a way to modify the swirl number, either by changing the injector outlet diameter D o , the conical end piece diameter C or the swirling vane SW, the minimum FTF gain value decreases as the swirl level increases. This is conrmed through 4 dierent set of experiments presented in Figs. 6.8, 6.10 and 6.12. One should however be careful that the swirl level is not the only parameter inuencing the minimum gain level. Two dierent ames, stabilized with the same swirl level but on two injectors featuring more than one dierent component, can lead to a dierent value of the gain minimum. This is highlighted in blue in Tab. 6.1, that summarizes the FTF results presented in this chapter. Moreover, if we consider two ames with dierent swirl numbers, the one with the higher swirl can lead to an higher minimum response if the two ames are stabilized on injectors featuring more than one dierent component. This is highlighted in red in Tab. 6.1.

These results are used to explain some of the dierences in the properties of the combustion instabilities analyzed in chapter 5.

In Fig. 5.11, it was found that the amplitude of the instability diers when the distance δ is set to 35 mm or to 50 mm and the length of the exhaust tube is modied from 1E to 2E and 3E. These self sustained oscillations were analyzed in a conguration with an injector equipped with the radial swirler SW 3 , a nozzle diameter D o = 20 mm and a conical end piece diameter C = 10 mm. The FTF of this injector is shown in Fig. 6.6-left. With δ = 35 mm, a strong instability was detected at a frequency close to 220 Hz with an exhaust length CONTENTS 109 of 1E. From Fig. 6.6-left, we observe that this frequency is close to the one where the FTF gain features a maximum value G 1. With δ = 50 mm, the strongest instability was detected at a frequency close to 164 Hz, with an exhaust tube length of 3E. Once again, this conguration features a maximum response with G 1 around this frequency in Fig. 6.6-left. With the same exhaust tube length 3E, the instability could be suppressed for the injector equipped with the same swirling vane but a smaller distance δ = 35 mm. One clearly sees that the FTF gain in Fig. 6.6-left is close to its minimum value G 0.4 at f = 164 Hz for δ = 35 mm.

In Fig. 5.13, it was observed that the intensity of the instability is close when the swirler design is modied but all the other geometrical parameters of the injector are xed. These self sustained oscillations were analyzed in a conguration with a nozzle diameter D o = 20 mm, a conical end piece diameter C = 10 mm and the swirling vanes SW 1 and SW 3 . Unfortunately the FTFs of these injectors were not measured. One can try to comment by analyzing FTF results obtained for SW 1 and SW 3 while all other geometrical parameters are xed as shown in Fig. 6.12. These data are obtained for a conguration featuring a nozzle diameter D o = 22 mm and a conical end piece diameter C = 14 mm, which is slightly dierent with respect to the one analyzed in Fig. 5.13 . One sees that the maximum gain value is G 1.5 for SW 1 and G 1.4 for SW 3 . These values are also observed in the same frequency range f = 160-170 Hz. It is thus evident that a change of the swirling vane when all the other parameters of the injector are xed does not lead to a sensible change of the maximum gain value or of the frequency at which this value is observed. FTF data for the same injectors studied in Fig. 5.13 are necessary to strengthen this conclusion.

110 Chapter -Effects of injector design and injection conditions on Flame Transfer Functions

Conclusion

Experiments were carried out to investigate changes in the acoustic response of swirl-stabilized lean premixed methane/air ames, when the system is operated at constant equivalence ratio φ = 0.82 for dierent injection conditions and injector designs.

It was rst shown that the ame response depends on the excitation frequency and amplitude. The transfer function measurements show that the FTF gain curve of swirling ames is characterized by a low frequency gain peak followed by a sudden drop and that the minimum gain value decreases as the swirl number increases. This was conrmed in dierent congurations obtained by modifying the diameter of the injection nozzle, the diameter of the conical end piece at the top of the central blu body and the dimensions of the radial swirler. On the other side, the FTF of a low-swirled ame was shown to be similar to that of a non-swirling V-ame, for which the gain exhibits values exceeding unity over a large frequency span before smoothly being reduced at higher frequencies.

It was shown that the behavior of the FTF gain with respect to the forcing level is highly non-linear at frequencies corresponding to maximum values of the gain, while it is roughly linear at the FTF gain minimum. The phase lag curve remains independent of the forcing level.

It was then found that a shift in the frequency range of the FTF gain and phase lag curves is obtained, when the distance between the swirler and the combustion chamber or the bulk ow velocity in the injection channel downstream the swirler are modied. This is due to a change of the time required for the convected disturbances generated at the swirler trailing edge to reach the burner outlet. It was observed that the product between the distance δ and the frequency f corresponding to a minimum ame response or the ratio between U b and f remains constant.

These results were used to interpret some of the features of combustion instabilities observed in the previous chapter. In particular, the fact that a maximum or a minimum value of the FTF gain can be obtained at a same frequency for dierent values of the distance δ between the swirler and the combustion chamber backplane, explains the dierent intensity of the instability observed at that same frequency for dierent values of the distance δ.

The following two chapters are dedicated to the interpretation of the results presented here by scrutinizing the ow and the ame with optical diagnostics and with the help of numerical simulations. The dierent strength of vortical structures shed from the burner rim is here not related to a dierent ame-ow interaction resulting from a dierent swirl number oscillations, since this eect is already observed in cold ow conditions and swirl number uctuations are comparable at the two frequencies. The origin of this mechanism is here purely hydrodynamic, resulting from the complex interaction between the axial and the azimuthal velocity components inside the injector and leading to a dierent shape of velocity proles at the burner rim.

Introduction

It has been emphasized in the introduction that the mechanisms leading to a drop of the FTF gain are not fully elucidated and dierent scenarios were envisaged to explain this drop. Palies et al. (Palies et al., 2010a;Palies et al., 2011e;[START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF] came to the conclusion that interferences between vortex shedding at the injector lip with a apping motion of the ame base do not let these vortices to grow in strength before they rollup the ame. [START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF] consider that the Kelvin-Helmholtz instability in the external shear layer of the forced jet cannot fully develop due to thermal dilation from the combustion reaction.

Results in the previous chapter showed that progressively increasing the swirl level led to a progressive reduction of the FTF minimum gain value. It has also been shown that the details of the injector geometry need to be considered as well. Dierent injectors featuring the same swirl level, led to slightly dierent reduction of the FTF gain.

The origin of the physical mechanisms leading to a drop of the FTF gain curve in the low frequency range needs further investigation. This is the main scope of this chapter. The following section is dedicated to the investigation of the forced ame dynamics by phase conditioned analysis of the OH* signal emitted by the pulsated ames obtained for dierent injectors. This is followed by phase conditioned analysis of PIV measurements made under cold ow conditions and by phase conditioned Large Eddy Simulations. These simulations are rst compared to PIV measurements and then further exploited to complete the analysis.

Flame dynamics

Four dierent injector geometries are considered for the analysis of ame dynamics. The corresponding ame transfer function measurements are reported in Fig. 7.1. The mean shape taken by these ames is exemplied in Figs. The OH * light distributions from the unperturbed ames are compared to the mean ame light distributions averaged over all phase conditioned images in the row. One can see that in all cases this average distribution remains essentially unaltered at the two forcing frequencies investigated, characterizing the minimum and maximum ame responses. In all forced congurations explored, the average light distribution over a forcing cycle also barely diers from the unperturbed ame light distribution, even for a forcing level as high as u /ū = 0.30 RMS. The main dierences when the ame is pulsated are (i) a small decrease of the mean ame lift-o distance, (ii) a small modication of the ame height and (iii) an increase of the ame volume in the region where vortical structures impinge the ame. This is emphasized by the green circle in the third sequence in Fig. 7.2-b.

Flame motion analysis

Further analysis is now made by determining the ame contours, highlighting the region of the ame where the luminosity is higher than a threshold level, for each image recorded. The threshold is taken here equal to 25% of the maximum ame luminosity in each phase averaged picture, a value found to be suitable for the determination of the ame contours. It was checked that the results presented hereafter are only weakly sensitive to this threshold level, when this is chosen in the range 15 -35% of the maximum ame luminosity. The ame contours are used to determine the ame height H f , which is deduced by calculating the vertical distance between the lowest point, shown as a plus sign in Figs. 7.2-7.3, and the highest point, shown as a diamond sign in Figs. 7.2-7.3, of the ame contours. The ame angle α with respect to the vertical direction is also determined at the ame leading edge position.

Finally, the swirl number S is also determined, for each selected phase in the forcing cycle. These data are deduced from the phase conditioned PIV elds measured in a longitudinal plane crossing the chamber axis and in a transverse plane 2 ± 0.5 mm above the top cone of the central blu-body in the absence of combustion. The precision on the deduced swirl number is ±5%. Evolutions of H f and α are plotted over two periods in Fig. 7.4 together with the evolution of the swirl number S.

The top graph in Fig. 7.4 is obtained for the injector with swirler SW 0 , an injector outlet diameter D o = 22 mm and a blu-body end piece diameter C = 14 mm at the forcing frequency f = 96 Hz. One reminds that in this case the ame features a weak swirl number S = 0.20 and that this case serves as a reference. The ame undergoes large uctuations of its height H f /H f , but no oscillation of the ame base angle α /α 0. The small oscillations of the swirl number S around the mean value 0.2 observed in this plot are meaningless here due to the low values of the swirl number.

The situation diers for the results obtained with swirler SW 3 in the other graphs in Fig. 7.4 independently of the forcing frequency and injector design.

The ame response now exhibits both ame base angle α /α oscillations and ame height H f /H f oscillations in the second and third rows in Fig. 7.4, obtained for a swirl number S = 0.80, but dierent injection nozzle diameters D o = 22 and 20 mm and top cone diameters C = 14 and 10 mm respectively. The same conclusions can be drawn for the results shown in the fourth row obtained for the same swirler SW 3 , but at a slightly lower swirl number S = 0.60 with D o = 18 mm and C = 12 mm. These swirled ames undergoing large oscillations of their ame base angle and large oscillations of their ame height all feature large oscillations of the swirl number in Fig. 7.4. These observations contrast with data reported by Palies et al. (Palies et al., 2010a;Palies et al., 2011e;[START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF] in which the ame base angle and swirl number oscillations diered at frequencies corresponding to maximum and minimum of the FTF gain.

The main striking features in Fig. 7.4 are that the oscillations of the swirl number S, the ame base angle oscillations α /α and the ame height oscillations H f /H f experienced by these ames, are very similar in the left and right plots corresponding the minimum and maximum ame responses respectively. The synchronization between these dierent signals is also roughly the same for the dierent cases explored. This conrms that there are no signicant differences between the motions executed by these ames, when they are forced at a frequency corresponding to a minimum or a maximum in the FTF gain curve. Another important nding is that large swirl number oscillations are detected for ames featuring low, but also large, FTF gain values. These two last observations contrast with current interpretations of ame responses to swirl oscillations (Palies et al., 2011e;[START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. They however need to be conrmed by further analysis.

Heat release rate uctuations analysis

The origin of the heat release uctuations observed at these excitation frequencies is now scrutinized by decomposing the heat release rate disturbances Q / Q as the sum of uctuations associated to changes of the volumetric rate of change of the heat released averaged over the ame volume < q > / < q > and uctuations associated to the ame volume V /V , with the methodology fully described in [START_REF] Gaudron | Impact of the References injector size on the transfer functions of premixed laminar conical ames[END_REF]:

Q Q = < q > < q > + V V (7.1)
The evolution of these three quantities is shown in Fig. 7.5 for the cases explored in Figs. 7.2-7.4. For the ame stabilized on the injector featuring SW 0 , D o = 22 mm and C = 14 mm leading to a low swirl number S = 0.20, heat release rate oscillations are mainly driven by changes of the ame volume Q / Q V /V , with only a slight contribution from local changes of the volumetric heat release rate in the top row in Fig. 7.5. In this case, the ame luminosity remains roughly constant along the ame arms over the forcing cycle in Fig. 7.2. In the second, third and fourth rows in Fig. 7.5, results are plotted for ames stabilized with the swirling vane SW 3 . These injectors feature high swirl numbers S ≥ 0.60. The contributions to heat release rate uctuations from V /V and < q > / < q > reach both comparable and relatively high levels in the right plots, when the ame response feature large FTF gain values. This means that large vortical structures impinging the ame modulate the ame shape but also change the local consumption rate. For the ames sharing the same swirl level S = 0.80, both signals V /V and < q > / < q > largely drop in the second and third rows in the left column in Fig. 7.5, for frequencies corresponding to low FTF gain values. At a slightly lower swirl level S = 0.60, there is no large drop of V /V and < q > / < q > and the level reached by these signals are comparable in the left and right graphs in the last row in Fig. 7.5 obtained at f = 110 and 180 Hz. In this latter case, one recalls that the FTF gain level remains high at these two frequencies as shown This analysis conrms the qualitative descriptions made in the discussion of Figs. 7.2-7.3. At the frequencies corresponding to the lowest ame responses, (i) the ame luminosity barely changes during the forcing cycle and (ii) the ame shape may execute a relatively large motion during the forcing cycle, but the overall surface and volume variation remains low. These two mechanisms explain the low FTF responses observed in this work. They also conrm that the reduced response of the heat release rate observed at specic frequencies is not associated here with some compensation mechanism between out of phase contributions to heat release rate oscillations associated to modications of the ame base angle dynamics with respect to the ame tip dynamics and resulting from dierent swirl number oscillations. They were shown to be the consequence of signicant reductions of both ame volume variations and local changes of the reaction rate compared to ames featuring a high response. It is now worth examining if these dierences could be interpreted with an analysis of the cold ow response.

Cold ow dynamics -PIV measurements

The investigation proceeds by examining phase conditioned PIV measurements with the acoustic pulsation gathered in an axial plane of the combustor in cold ow conditions. The PIV setup is described in chapter 2. To identify vortical structures, the Q criterion [START_REF] Jeong | On the identication of a vortex[END_REF] is selected:

Q = 1 2 (|Ω| 2 -|S| 2 ) (7.2)
where S and Ω are the symmetric and anti-symmetric components of the velocity gradient respectively. Iso-contours of Q are inferred from velocity measurements conducted in the axial plane crossing the center of the combustion chamber. Only positive values of Q are retained, while negative values, indicating areas of the ow where shear is present but no swirling motion, are forced to a zero value. As the velocity eld is only known in the axial plane, Q is here calculated on a 2D basis:

Q = ∂u z ∂z ∂u x ∂x - ∂u x ∂z ∂u z ∂x . (7.3)
where u x and u z are the transverse and axial velocity components. This approximation of course does not reproduce the full complexity of the swirled 3D ow response to the ow modulation, but it is reasonable because the analysis of the phase averaged ame response in Figs. 7.2-7.3 revealed that their dynamics also diers in the axial plane.

Results are presented in Fig. 7.6 and Fig. 7.7 for the same injectors as in Fig. 7.2 and 7.3 respectively, except that combustion is not initiated. Vortical structures are detected both in the external and internal shear layers of the jet. The structures in the internal shear layer can however not develop in reacting ow conditions because the ame leading edge is anchored on the central blu body. This is why the following analysis only considers the response of the external shear layer to acoustic forcing.

The rst and third sequences in Fig. 7.6 reveal that large coherent vortical structures are shed within the cold ow from the external lips of the burner at the forcing frequencies f = 96 Hz and f = 170 Hz when the same injector (D o = 22 mm and C = 14 mm) is equipped with swirler SW 0 (S = 0.20) and SW 3 (S = 0.80) respectively. These excitation conditions correspond to FTF gain maxima in Fig. 7.1. In both cases, a large vortical structure is generated at the rim of the injector and is convected downstream in the chamber. This is highlighted by the high values taken by Q in these gures that exceeds Q > 2. 10 6 s -2 . The displacement of the center of these structures is emphasized by the superposed dashed lines. The same observations can be made for the results shown in the second and fourth rows in Fig. 7.7 obtained for an injector equipped with swirler SW 3 but dierent nozzle and top cone diameters. The forcing conditions in the second row at f = 190 Hz and in the fourth row at f = 180 Hz are associated to a maximum of the FTF gain plots in Fig. 7.1. These data reveal that large coherent vortical structures shed in the external shear layer of the ow are responsible for the high gain values taken by the FTF gain independently of the injector design.

Results are now analyzed for excitation conditions leading to minimum FTF gain values. The second sequence in Fig. 7.6 and the rst in Fig. 7.7 correspond to the response of the ow for the injector with the swirler SW 3 but dierent nozzle and top cone diameters leading to the same swirl level S = 0.80. These two ames feature a minimum of the FTF gain in Fig. 7.1 at f = 96 Hz and 110 Hz respectively. In both cases, the large coherent structures shed from the burner lip in the external shear layer can hardly be identied with Q values lower than Q < 1. 10 6 s -2 . These weak structures are also rapidly dissipated by the ow as they are convected downstream.

When the swirl level is slightly decreased to S = 0.60 for an injector with the same swirler SW 3 , but a reduced nozzle diameter D o = 18 mm and a top cone diameter C = 12 mm, one now again may identify a weak vortical structure shed from the burner lip and travelling in the external shear layer of the ow in the third sequence in Fig. 7.7 for an excitation frequency f = 110 Hz. The strength of this structure with Q < 1.5. 10 6 s -2 is however much lower at this forcing frequency than that observed for the same injector when the ow is forced at f = 180 Hz in the last sequence in Fig. 7.7 for which Q > 2.5. 10 6 s -2 . One Interactions of the ame with large vortical structures shed from the burner rim is a well known mechanism leading to nonlinearities in the ame response [START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF][START_REF] Thumuluru | Characterization of acoustically forced swirl ame dynamics[END_REF]Oberleithner et al., 2015). In Fig. 6.3, it was shown that nonlinearities are strong at the FTF gain maximum, but are drastically reduced at the FTF gain minimum. indicate that this behavior is linked to the lower strength of the vortical structures shed from the burner rim at forcing frequencies corresponding to minimum FTF gain values.

Reduction of the strength of large vortical structures at the frequency corresponding to the FTF gain minimum was already identied in previous works on swirling ame dynamics (Palies et al., 2011e;[START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. In these studies, however, the origin of this reduction was in both cases attributed to some interference mechanism between the ame dynamics and the forced shear layer. obtained in cold ow conditions reveal that these vortical structures are also damped even in the absence of heat release. This in turn leads to the conclusion that the strong reduction of the FTF of swirled ames anchored on a central blu body at peculiar frequencies is a purely hydrodynamic mechanism, which nds its origin from the complex interactions taking place between acoustic waves and the vorticity eld of the swirled ow dynamics. This is why this problem is further investigated with the help of LES calculations of the pulsed cold ow. The objective of these simulations is to better highlight the mechanisms leading to a reduction of the strength of vortical structures at the forcing frequencies corresponding to minimum FTF gain levels. To do so, Large Eddy Simulations were carried out on the conguration described in Chapter 3 for the injector equipped with the swirler SW 3 , a nozzle injection diameter D o = 20 mm and a blu body with a conical top cone diameter C = 10 mm. One recalls that this conguration is slightly dierent from those presented so far, since it features a 1 mm shorter central rod. The ow is forced at two frequencies, f = 120 Hz and f = 190 Hz, respectively corresponding to a FTF minimum and maximum gain value in Fig. 6.3. The question arises whether the observations highlighted by PIV in the previous section originate from a modication of the response of the swirled ow to acoustic forcing outside the burner or if this could already be explained by the dynamics of the ow within the burner.

Figure 7.8 highlights some cross sections inside the injector that are used in this chapter to analyze the data obtained from LES. Two of these sections are located upstream the swirler unit. Section HW corresponds to the one where the hot wire probe is inserted within the ow in the experiments. Section S-E, corresponds to the sudden expansion of the cross section between the 22 mm diameter tube where the hot wire is inserted and the enlargement of 40 mm diameter necessary to mount the swirler unit. The other sections are located downstream the swirling vane SW 3 . Section AS 08mm located in the injection nozzle at a distance of 8 mm downstream the swirler backplane. Section AS 56mm corresponds to the injector outlet section which is ush mounted to the combustion chamber backplane.

S-E HW

AS 08mm AS 56mm inlet Acoustic forcing is introduced in the numerical simulations at the inlet section in Fig. 7.8 through the INLET_RELAX_UVW_T_Y_PULSE boundary condition from the AVBP solver. This boundary condition allows to superpose a harmonic acoustic uctuation of desired frequency and amplitude to the mean ow. Acoustic excitation is introduced in the simulation by starting from a converged solution for the ow eld in the absence of acoustic forcing as described in chapter 3. Simulations are then carried out until the transient response of the ow is evacuated for several forcing periods, typically more than 10, before comparing the simulations and experiments.

It is important to underline that all the data presented at the dierent locations illustrated in Fig. 7.8 are obtained after a spatial integration over the cross section of interest. For example the axial velocity shown in Fig. 7.9-left is obtained as:

u z (t) = 1 A 2πru z (r, t)dr. (7.4)
The temporal evolution of the velocity and acoustic pressure signals obtained with LES are compared to hot wire and microphone measurements in the HW section in Fig. 7.9. It is worth recalling that the ow is laminar in this section at the outlet of the convergent unit with a top hat velocity prole. Five cycles of excitation are shown. Results are presented at the top in Fig. 7.9 for f = 120 Hz corresponding to a minimum FTF gain as shown in Fig. 6.3. The response at f = 190 Hz at the bottom in Fig. 7.9 corresponds to a maximum FTF gain. In both cases, the measured and simulated velocity signals are well superposed in the left graphs in Fig. 7.9.

The corresponding acoustic pressure signals in the right plots are a bit overestimated by the LES. This is due to the diculty to adjust the correct acoustic forcing level at the inlet of the numerical domain. One cannot prescribe the desired acoustic velocity and desired acoustic pressure wave in section HW in the numerical solver. One prescribes the forcing frequency and amplitude of the Riemann invariant at the numerical domain inlet, which corresponds to the amplitude of the characteristic wave entering the numerical domain, while the characteristic wave leaving the domain through the inlet is adjusted with a relax coecient [START_REF] Selle | Actual Impedance of Nonreecting Boundary Conditions: Implications for Computation of Resonators[END_REF]. One then needs to correctly adjust the value of the relax coecient with an iterative procedure to get the desired level on both the acoustic velocity and acoustic pressure in section HW. This procedure is a crude way to mimic the impedance of the real system in the section HW.

Overall, we consider that the comparisons in Fig. 7.9 are satisfactory for the sake of the analysis to be conducted.

The next step, after having checked that the correct acoustic perturbation is introduced in the computational domain at the two frequencies of interest, is to compute the Q criterion contours deduced from LES to get phase averaged results with respect to the excitation signal. This is done in a two steps process. Two types of averaging are performed. A phase-average procedure is applied over 25 cycles of excitation at both excitation frequencies. This procedure is applied for 24 phases in one acoustic cycle separated by 15 o . The phase reference for averaging is the velocity signal in the HW section. To further improve the results, an azimuthal average is then performed over 10 equally separated axial cuts crossing the burner axis with dierent azimuths. Two cuts at azimuths separated by 90 o are illustrated in Fig. 7.10. All these data are projected to the x -z plane. The resulting scan of LES data are post-processed to calculate the two dimensional Q criterion contours with Eq. ( 7.3) as done for the PIV data.

Results of this procedure are shown in Fig. 7.11 for 6 phases in the forcing cycle. Only the Q criterion contours inside the chamber are shown. It is evident that vortical structures are much stronger in the bottom sequence in Fig. 7.11 for the forcing frequency f = 190 Hz corresponding to a maximum FTF gain value, than for the sequence of images shown at the top for f = 120 Hz corresponding to a minimum FTF gain value. At both frequencies, the formation of a coherent vortex starts at the external rim of the injector at a phase between 300 o and 0 o with respect to the velocity signal in section HW. This vortical structure is then convected through the combustion chamber along the shear layer of the swirled jet and it is eventually dissipated. Its trajectory is represented by the dashed-dotted blue line. Figure 7.11 shows that the structure is dissipated much faster at the forcing frequency f = 120 Hz, corresponding to a minimum FTF gain value than at f = 190 Hz corresponding to a maximum FTF gain.

Discussion

One may rst wonder if this behavior may be linked to some preferential Strouhal number. A number of studies indicate that staggered vortical structures are released in the shear layer of an acoustically forced jet or in the wake ow downstream a blu-body at a preferential characteristic frequency given by:

f = St ū D (7.5)
where ū is the bulk ow velocity at the injection nozzle outlet, D is the outlet diameter of the injector and the Strouhal number St 0.20 -0.30 independently of the Reynolds number [START_REF] Crow | Orderly structure in jet turbu-lence[END_REF][START_REF] Cantwell | An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder[END_REF][START_REF] Prasad | The insability of the shear layer separating from a blu body[END_REF][START_REF] Emerson | Density ratio eects on reacting blu-body ow eld characteristics[END_REF]. These studies however do not deal with the response of swirling ows. Nonetheless, it is worth estimating the Strouhal numbers in the case explored in this study. Taking the At this stage, it is worth taking a step back to the FTF measurements presented in chapter 6. One focus on FTF results shown in Fig. 6.6-left obtained for the same injector and ow conditions but dierent values of the distance δ between the swirling vane and the combustion chamber backplane. The conguration with the swirler set at δ = 50 mm, features a maximum gain value at f = 190 Hz. One could argue that this is related to a preferential Strouhal number, for example St = 0.24 leading to the largest coherent structures released within the ow. A change in the distance δ does however not alter any value in Eq. ( 7.5), while in Fig. 6.6-left, the FTF gain at the same forcing frequency, and thus the same Strouhal number, corresponds to a minimum value when the distance is set to δ = 35 mm.

The same conclusion can be drawn from the analysis of results presented in Fig. 6.12. In these cases, the swirler design was modied, but the other geometrical parameters were xed so that the velocity ū and the hydraulic diameter D remain unchanged in Eq. (7.5). Nonetheless, at a same frequency f 100 Hz, maximum and minimum gain values are observed for the FTF obtained with the dierent swirlers tested. show that maximum and minimum gain values are respectively linked to high and low strengths of vortical structures released in the ow. This strength cannot be related to a preferential Strouhal number since for a same Strouhal number St, low and high FTF gain value are observed in Fig. 6.6-left and 6.12.

Analysis of signals inside the injector unit

The Large Eddy Simulations of the cold ow response inside the combustor conrm the results obtained by PIV discussed in the previous section. LES signals are now analyzed inside the injector in a region where experimental measurements could not be achieved. Figure 7.12 presents the temporal evolution of axial and azimuthal velocity components and pressure signals in sections HW, AS 08mm and AS 56mm . At a rst glance, it is hard to nd any dierence between the results reported on the left obtained for f = 120 Hz (corresponding to a FTF minimum gain) and those plotted on the right for f = 190 Hz (maximum FTF gain). The axial velocity signals, in red, are the same at the two frequencies in section HW . This was already illustrated in Fig. 7.9. Even downstream the swirler, in section AS 08mm , the axial velocity signals remain unaltered in phase and amplitude by the swirler at the two excitation frequencies. The only small dierence is seen in section AS 56mm where the average axial velocity u z increases due to the ow passage constriction shown in Fig. 7.8 at the injector outlet. This acceleration however is the same in both cases and does not change the phase lag of these axial velocity signals. One may also conclude that ow dissipation does not alter axial velocity disturbances at f = 120 and 190 Hz. These confrontations conrm the acoustic nature of the axial forcing as already discussed in Palies et al., 2011a.

The ow is purely axial in section HW in Fig. 7.12. The swirler introduces a certain amount of rotation in the ow. The average azimuthal velocity component conferred to the ow is roughly the same at both forcing frequencies f = 120 Hz and f = 190 Hz in section AS 08mm . In particular there is no phase lag between the left and right plots in Fig. 7.12. Contrarily to results found for the axial velocity disturbances, azimuthal disturbances are higher in section AS 08mm than at the outlet of the burner in section AS 56mm due to dissipation and this is true at both forcing frequencies. One may also see that the phase lag of the azimuthal with respect to the axial component of velocity also diers between sections AS 08mm and AS 56mm and diers in the same section AS 56mm between results found at f = 120 Hz and f = 190 Hz.

Finally, Fig. 7.12 also shows that the mean pressure p and pressure amplitude p drop between section HW and the injector outlet. Their phase lag in the left and right plots obtained for f = 120 Hz and f = 190 Hz remains unaltered through the swirler, but their amplitude drops signicantly.

The main dierence between data shown in Fig. 7.12 at f = 120 Hz and f = 190 Hz originate from the phase-lag ∆ϕ u θ -u z between the axial u z and azimuthal u θ velocity signals. These signals are almost in phase in section AS 08mm , but their phase lags dier in section AS 56mm . At this location, u z and u θ are still almost in phase at f = 190 Hz, while they are close to out of phase for the excitation at f = 120 Hz.

The phase lag between the axial or the azimuthal velocity component in different sections downstream the swirler outlet with respect to the axial velocity signal in section HW is determined. Results are shown in Fig. 7.13-left. The phase-lag ∆ϕ u z -u z between the axial components u z in section HW and the u z signals at the dierent AS locations is rst considered. This phase-lag is always close to zero at all locations downstream the swirler backplane. This conrms that axial velocity disturbances propagate at the speed of sound through the swirler and injection tube up to the burner outlet. The distance l covered by the acoustic waves between the swirler backplane and the burner outlet remains small with respect to their wavelengths at f = and 190 Hz, He = ωc/l 1. One can neglect this contribution to the overall phase lag and consider that axial disturbances remain in phase through the injector.

One can also see in Fig. 7.13-left that ∆ϕ u θ -u z corresponding to the phase lag of the azimuthal velocity u θ in the dierent AS locations downstream the swirler backplane with respect to the axial velocity in section HW increases roughly linearly with the frequency of perturbation. Considering that azimuthal ow disturbances are transported by the mean ow, the phase lag ∆ϕ u θ -u z can be expressed as [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Palies et al., 2011a):

∆ϕ u θ -u z = 2πf δ u c (7.6)
where u c is the convective velocity of azimuthal perturbations and δ is the distance from the swirler exit where azimuthal perturbations are generated.

Taking the slope of the curve in Fig. 7.13-left yields the velocity u c which is plotted in Fig. 7.13-right. It is either normalized by the bulk ow velocity u in the corresponding cross section or the maximum velocity u m reached by the ow in the section of interest. It is interesting to observe that u c lies in between u and u m , but is not modied by the forcing frequency f . The phase lag dierence observed at f = 120 Hz and f = 190 Hz between azimuthal perturbations in Fig. 7.12 at the burner outlet in section AS 56mm is not due to changes of the phase speed at which these disturbances travel.

These velocity signals are further analyzed in Fig. 7.14 by considering their relative uctuations to deduce the resulting swirl number uctuations S with:

S S = u θ u θ - u z u z (7.7)
This linearized expression for perturbations of the swirl number was used by Palies et al. (Palies et al., 2010a;Palies et al., 2011e) to show that, due to the dierent phase lags between u z and u θ signals with frequency, the largest swirl number oscillations S are obtained when the axial and azimuthal velocity components are out of phase at the burner outlet. Conversely, the smallest swirl number oscillations correspond to situations in which the axial and azimuthal velocity components are in phase at the burner outlet. They used this model to show that the amplitude of the oscillations of the ame angle at the ame base increases with the amplitude of the swirl number oscillations. They also found that at a frequency corresponding to a minimum FTF gain, the swirl number oscillations were the largest, while they were considerably reduced at forcing frequencies corresponding to a maximum FTF gain. This in turn was used to develop a scenario in which strong interactions between a apping ame and vortices shed in the shear layer from the injector rim led to a reduction of the strength of the shed vortical structures.

It has already been shown in Fig. 7.4 that swirl number oscillations S reach comparable levels at the two frequencies f = 120 and 190 Hz respectively corresponding to a minimum FTF gain and a maximum FTF gain. It was also shown that the strength of these vortical structures is reduced at f = 120 Hz compared to those released at f = 190 Hz even in the absence of combustion.

It is now found that the swirl number uctuations deduced from Eq. (7.7) are slightly higher at f = 120 Hz than at f = 190 Hz in Fig. 7.14, but they reach in both cases quite high levels. One hardly sees how the slight dierent levels observed in Fig. 7.14 for S could explain the large dierences in vortex strength highlighted in Fig. 7.11.

One possibility is that Eq. (7. 7) is an oversimplied model that does not well reproduce the swirl number disturbances seen by the ow at the burner outlet.

The swirl number can also be deduced from the original denition Eq. (5.1). It is here deduced from the axial and azimuthal velocity proles at the outlet of the burner in section AS 56mm for 24 phases in the forcing cycle from the axial cuts shown in Fig. 7.10. Results are superposed in Fig. 7.14. The values of S /S deduced from the velocity proles and calculated with the exact expres- sion Eq. ( 5.1) or those deduced from the simplied linearized model Eq. (7.7) are close in Fig. 7.14. This means that the simplied expression Eq. ( 7.7) is well suited to get an estimate of the swirl uctuation level. The value of S deduced from the exact expression Eq. ( 5.1) is now slightly higher at f = 190 Hz than at f = 120 Hz. More importantly, it is once again shown that the swirl number uctuations are strong at both forcing frequencies. This conrms that the large dierences observed for the vortical structures shed at these two frequencies in the cold ow do not directly result from changes of the level of swirl number uctuations.

Vortex formation without swirl

To get more insight, the mechanisms leading to shedding of large coherent structures with and without swirl are now examined. The formation of vortical structures in a purely axial ow is analyzed rst at section S-E upstream the swirler corresponding to a sudden expansion of the cross ow section passage in Fig. 7.8. Results post-processed with the same procedure already described for the analysis of the data shown in Fig. 7.11 are presented in Fig. 7.15.

This gure reveals that the strength of vortical structures shed at an excitation frequency f = 120 Hz and f = 190 Hz in the absence of swirl is the same. In both cases, a large vortex is formed between phases 300 o and 0 o and then fully detaches from the rim between phases 60 o and 120 o . The vorticity and the Q criterion reach about the same values for both results presented at f = 120 Hz and f = 190 Hz. This test highlights without any ambiguity that the two dierent responses of the injector at f = 120 Hz and f = 190 Hz is linked to the dynamics downstream the swirling vane.

Velocity proles in section S-E are also analyzed in Fig. 7.16 as a function of their phase lag with respect to the acoustic velocity in section HW. Without surprise, the axial proles coincide in the left and right top graphs obtained for the two forcing frequencies f = 120 Hz and f = 190 Hz respectively. An attempt is also made to explore the dynamics leading to the detachment of these vortical structures with respect to changes of the axial velocity prole at the channel wall. The gradient du z /dr is plotted in Fig. 7.16. This velocity gradient, obtained for 0.9 ≤ r/R ≤ 1, is proportional to the shear on the channel wall. One again sees that the shear wall follows exactly the same law for the two excited ows when the signal is plotted as a function of the phase in the oscillation cycle. As the axial velocity increases for phases between 270 o to 90 o , a vortex is rst generated at the channel outlet and then detaches. In the following part of the forcing cycle, this structure is convected along the dashed-dotted blue line in Fig. 7.15.

Another attempt is made to characterize the strength of the shed vortical structures. This is done by measuring their circulation Γ along their trajectory, around the boundary of a closed surface surrounding and following the struc-ture. The circulation Γ corresponds to the ux of vorticity through this surface.

Only the vorticity linked to the vortical structure needs to be considered. For this purpose, a binarization procedure is applied: (i) a threshold is selected on the Q criterion contours; (ii) a binary matrix is constructed by assigning 1 for all values above this threshold and assigning 0 for all values below; (iii) the vorticity is multiplied by this binary matrix. The vorticity scan resulting from this procedure is shown in Fig. 7.15, on the left of the symmetry axis. The surface used to obtain the circulation is superposed on the images as well.

The results in the bottom right plot in Fig. The same observations are made for the excitation frequency f = 120 Hz. The only dierence is that Γ saturates at a higher level and then slightly drops. This gure conrms that the strength and the circulation of the shed vortical structures at f = 120 Hz and f = 190 Hz are very close in the absence of swirl.

It was also checked that these results are only weakly sensitive to the selected threshold level to determine the circulation.

Vortex formation with swirl

The same analysis is repeated at the outlet of the injector in the cross section AS 56mm . In this section, the ow is swirled with a strong azimuthal velocity component. It was already shown in Fig. 7.11 that vortical structures shed from the burner lip are stronger at an excitation frequency f = 190 Hz compared to those produced at f = 120 Hz. Further insight into the mechanisms leading to this behavior of the swirled ow response can be gained by analyzing the velocity proles shown in Figs. 7.17 and 7.18.

In Fig. 7.17, the axial velocity proles superposed at regularly distributed instants in the forcing cycle show that the envelope of the axial velocity close to the central blu body is wider at f = 120 Hz in the left plot than in the right plot obtained for a forcing frequency f = 190 Hz. The situation is reverse for the envelope covered by the axial velocity close to the solid wall of the injection channel. The peak axial velocity close to the channel wall is bounded between 9 and 16 m.s -1 in the left plot obtained for an excitation at f = 120 Hz. The peak velocity in the boundary layer between 0.8 < r/R o < 1 spans from 7.5 to 17.5 m.s -1 when the ow is excited at f = 190 Hz in the right plot. As a consequence, the axial velocity prole is steeper at f = 190 Hz close to the channel walls when the ow is excited at f = 190 Hz and larger changes of the radial velocity close to the central injector when the ow is excited at f = 120 Hz.

The minimum axial velocity -3 m.s -1 reached by the ow at a phase 270 o for an excitation at f = 120 Hz is much lower in Fig. 7.17 compared to the value -1 m.s -1 found for the axial ow velocity when the ow is excited at f = 190 Hz. This may explain the higher propensity of ashback during FTF measurements when the forcing frequency is set close to f = 120 Hz as discussed in chapter 6.

As the velocity prole features larger variations close to the solid wall of the injection channel at f = 190 Hz than at f = 120 Hz, this leads to larger wall gradients du z /dr (determined at a distance of 1 mm from the rim) in the bottom left plot in Fig. 7.17 when the ow is excited at f = 190 Hz. There is no phase lag between excitation at f = 120 and 190 Hz. The only dierence is the amplitude of the oscillation experienced by the shear at the wall.

Finally, the circulation Γ around a contour shown in Fig. 7.11 following the vortex convected by the ow is extracted from the simulations with the same method described for results in Fig. 7.16. The circulation Γ in the bottom right plot in Fig. 7.17 reaches values of about 250 m 2 .s -1 at f = 190 Hz compared to the ow forced at f = 120 Hz for which circulation is limited to 150 m 2 .s -1 . The initial growth of circulation is the same for f = 120 and 190 Hz for phases lower than 70 o . Saturation and dissipation then alters the circulation of the vortical structure shed at f = 120 Hz, while the circulation of the vortical structure continues to grow for the ow forced at f = 190 Hz until phases of about 90-100 o . This conrms that after the initial phase of vortex formation and growth, dissipation of the coherent structure circulation is much faster at the frequency f = 120 Hz corresponding to a minimum FTF gain value than at f = 190 Hz corresponding to a maximum FTF gain value.

Conclusion

Experimental measurements and numerical simulations have been carried out to investigate the ame and ow dynamics above a swirled injector at frequencies f = 190 Hz and f = 120 Hz leading to maximum and minimum values of the FTF gain as shown in the previous chapter.

Phase conditioned OH* chemiluminescence images revealed that the ame motion is qualitatively similar at these two frequencies. For ames stabilized by a central blu body with a vanishingly small level of swirl (S = 0.20), the ame height uctuation is dominant with respect to ame angle uctuations, while as the swirl number increases (S > 0.60) both contributions are important. The dierence in ame dynamics between low and high values of the FTF gain has been analyzed by comparing the ame luminosity uctuations and ame volume uctuations. A general decrease of these two contributions to the heat release rate oscillation has been observed as the gain level is reduced.

Phase conditioned PIV measurements have then been considered to analyze the cold ow response to these acoustic excitations. These data conrmed that vortical structures shed from the burner rim are stronger at the frequency of the FTF gain maximum, as reported in the literature. They also revealed that these vortical structures vanish at forcing frequencies corresponding to a minimum value of the FTF gain. The originality lies here in the fact that these data were obtained in cold ow conditions, while interaction of the ame with the ow dynamics was so far believed to be responsible of this phenomenon.

Phase conditioned numerical simulations have lastly been conducted in cold ow conditions at the two excitation frequencies respectively leading to a maximum and a minimum value of the FTF gain. These data rst conrmed the CONTENTS 145

PIV results in terms of dierent strength of vortical structures shed from the burner rim. The velocity and pressure signals inside the injector, at locations where experimental measurements were not available, have then been analyzed.

No important dierences could be observed in this internal response of the injector except in the phase lag between the axial and the azimuthal velocity proles at the burner outlet. In particular, the swirl number uctuations, often considered to be a discriminating parameter for the dierent behaviors of a swirled injector at dierent frequencies, was shown to reach comparable level of oscillation in these simulations at the two frequencies of interest.

The higher strength of the vortical structures shed from the burner lip and convected in the combustion chamber at the frequency corresponding to the maximum FTF gain level, has been here found to be related to a dierent shape of the envelope of the axial and radial velocity proles at the burner outlet. The ow is submitted to a much higher shear du/dr at the channel wall at f = 190 Hz corresponding to a FTF gain maximum than when the ow is forced at f = 120 Hz. The envelope of the radial velocity prole also signicantly diers between the two excitation conditions, with much larger changes close to the central blu body at f = 120 Hz compared to results obtained at f = 190 Hz.

These observations lead to the conclusion that the mechanism leading to high and low gain values of the FTF of swirled ames stabilized by a blu body is purely hydrodynamic. Further eort needs to be made to model this behavior.

Chapter 8

Flame Transfer Function scaling

Introduction

The determination of the parameters describing the ame transfer function (FTF) with a reduced set of dimensionless numbers is an important issue for engineers. This type of scaling law allows to reduce the set of experiments or simulations needed to determine the FTF for dierent ow operating conditions. An attempt is made in this study to nd such a scaling law when the ow operating conditions and the burner design are modied.

In the cases of laminar premixed conical and V-ames submitted to owrate perturbations, this issue has been extensively covered [START_REF] Merk | An analysis of unstable combustion of premixed gases[END_REF][START_REF] Fleil | Response of a laminar premixed ame to ow oscillations: A kinematic model and thermoacoustic instability results[END_REF][START_REF] Ducruix | Theoretical and experimental determinations of the transfer function of a laminar premixed ame[END_REF]. These studies came out with two main dimensionless numbers that were shown to govern the ame frequency response to incoming owrate disturbances [START_REF] Schuller | A unied model for the prediction of laminar ame transfer functions: comparisons between conical and V-ame dynamics[END_REF]. For a ame with a laminar burning velocity S L stabilized over a burner of radius R with a bulk ow velocity U b and submitted to small owrate perturbations at angular frequency ω, the two main independent dimensionless numbers controlling its response are (i) a reduced frequency ω * = ωR/(S L cos α), where sin α = S L /U b yields the mean position of the ame sheet, and the ratio of laminar burning to the In the case of premixed swirling ames submitted to incoming owrate disturbances in constant equivalence ratio mixtures, a Strouhal number based on the ratio of a ame length scale to a convective velocity is generally used to adimensionalize the FTF results [START_REF] Kulsheimer | Combustion Dynamics of Turbulent Swirling Flames[END_REF]Kim et al., 2010a;Kim et al., 2010b;Palies et al., 2010b;[START_REF] Jones | Flame Response Mechanisms Due to Velocity Perturbations in a Lean Premixed Gas Turbine Combustor[END_REF] :

St = f L f u vc (8.1)
where L f is a ame length scale and u vc is an eective convection velocity. Kim et al., 2010a used this scaling law to reduce their FTF data gathered for premixed swirled ames at equivalence ratios varying from φ = 0.65 to 0.75 and a xed bulk ow velocity U b = 30 m/s on the same curve with a reasonable match. The distance L f was dened as the center of mass of CH* emission images recorded over one forcing period and was found to be a function of the equivalence ratio and the forcing frequency. The eective convective velocity was determined as u vc = L f /τ , where τ is deduced from the slope of the FTF phase lag. This type of scaling law can only be used once the FTF is determined and does not consider the impact of the swirler.

Many recent investigations of the FTF of swirling ames by modeling tools [START_REF] Hirsch | Inuence of the swirler design on the ame transfer function of premixed ames[END_REF][START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF][START_REF] Acharya | Swirl effects on harmonically excited, premixed ame kinematics[END_REF], numerical simulations (Palies et al., 2011e;[START_REF] Acharya | Eect of azimuthal ow uctuations on ow and ame dynamics of axisymmetric swirling ames[END_REF] and by experimental means [START_REF] Huang | Dynamics and stability of lean-premixed swirl-stabilized combustion[END_REF]Palies et al., 2010b;[START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF], indicate that the injector dynamics and the response of the swirler to ow perturbations need to be considered to interpret the key features of the FTF of swirling ames. These studies revealed that an acoustic pulsation impinging on a swirler generates a vorticity wave which is then convected downstream at the local ow velocity. This in turn leads to interferences between axial acoustic and azimuthal convective disturbances, along the distance from the swirler exit to the combustion chamber inlet (Palies et al., 2010b;[START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Palies et al., 2011a). Straub and Richards, 1999 rstly described the inuence of the axial swirler position on the self-excited instabilities in a gas turbine combustor. Komarek and Polifke, 2010 and later Kim and Santavicca, 2013b observed a strong dependence of the Flame Transfer Function (FTF) upon swirler-combustor distance.

In a series of works dedicated to understanding the key mechanisms controlling the dynamics of swirling ames, Palies and co-workers [START_REF] Palies | Experimental study on the eect of swirler geometry and swirl number on ame describing functions[END_REF]Palies et al., 2010b;Palies et al., 2011e) found that (i) the same mechanisms take place for radial or axial swirlers, (ii) the same FTF shape is obtained when velocity uctuations are measured on the upstream or downstream side of the swirler and (iii) the frequencies at which the lowest FTF gain values are ob- , which may also be rewritten as :

St = f (L f + δ) U b (8.2)
Note that only the bulk ow velocity U b is involved in this expression instead of a convective velocity u vc .

This problem is revisited here. One takes advantage of the results shown in the previous chapters to sum up the main physical mechanisms leading to the peculiar shape of the FTF of swirling ames.

Mechanisms controlling the FTF of swirling ames

In the following analysis, two dominant mechanisms, illustrated in the sketch in Fig. 8.1, are considered to be at the origin of the response of lean premixed swirling ames submitted to acoustic perturbations.

The rst mechanism is associated to the interaction between acoustic and convective disturbances. When an acoustic wave, with acoustic velocity u z , impinges on a swirler, two waves are generated on the downstream side of the swirler [START_REF] Wang | Unsteady ow evolution in swirl injectors with radial entry[END_REF]Palies et al., 2010b;[START_REF] Hirsch | Inuence of the swirler design on the ame transfer function of premixed ames[END_REF][START_REF] Kim | Interference mechanisms of acoustic / convective disturbances in a swirl-stabilized lean-premixed combustor[END_REF]. They correspond to a transmitted acoustic wave, with the same acoustic velocity u z if the swirler is transparent to acoustic waves in the absence of a large pressure drop, and a vorticity wave, with azimuthal velocity u θ that is generated at the swirler trailing edge and is convected by the mean ow. These waves travel at dierent speeds inside the injector from the swirler exit up to the combustion chamber backplane. Their phase lag can be expressed as:

ϕ u θ -u z = 2πf δ 1 u c - 1 c 2πf δ u c (8.3)
where u c is the velocity at which azimuthal velocity uctuations are convected inside the injector and c u c is the speed of sound. This phase lag controls the frequency f at which the FTF gain reaches a minimum (Palies et al., 2010b;Palies et al., 2011e;[START_REF] Kim | Interference mechanisms of acoustic / convective disturbances in a swirl-stabilized lean-premixed combustor[END_REF]. In particular, it was shown in Palies et al., 2010b that the frequency corresponding to the lowest FTF gain value is associated to out of phase azimuthal and axial ow disturbances at the injector outlet.

The second mechanisms controlling the FTF is ame-vortex interaction [START_REF] Kulsheimer | Combustion Dynamics of Turbulent Swirling Flames[END_REF][START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF][START_REF] Durox | Experimental analysis of nonlinear ame transfer functions for dierent ame geometries[END_REF]Oberleithner et al., 2015). Vortices are generated at the rim of the injector, convected in the swirled jet external shear layer through the combustion chamber and interact with the ame. In Fig. 8.1, a same vortex is represented at three instants during a forcing cycle. The delay τ , appearing in the FTF phase lag, ϕ = 2πf τ , can be expressed as the ratio of the convective velocity of these structures, u vc , to the distance they cover before interacting with the ame, L CG .

Scaling of the transfer functions of swirling ames

Attempt is rst made to scale the FTF data with: 8.4) where L CG is a characteristic ame dimension, which is here taken equal to the distance between the injector rim and the ame center of gravity (Kim and Santavicca, 2013a;[START_REF] Osi¢ | Response of a swirl-stabilized ame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF] and u vc is the convective velocity of vortical structures inside the combustion chamber. The coordinates of the x cg = i,j x i,j y i,j I OH * ,i,j i,j y i,j I OH * ,i,j

St a = f L CG u vc (
; y cg = i,j y i,j y i,j I OH * ,i,j i,j y i,j I OH * ,i,j ,

where i and j are pixel indexes, x i,j and y i,j the pixel coordinates and I OH * ,i,j the measured OH* intensity. The distance L CG is obtained from Eq. ( 8 cycle around a mean value that increases when the ow passage cross section, area A 2 , decreases as indicated in Table 4.1. One can also see that u vc drops in the nal portion of the cycle. This could be due to the fact that it is more dicult to discern vortices from the Q criterion plots shown in Fig. 7.6 for the latest phases, since the vortex intensity is dissipated during its convection in the chamber. Nonetheless, an average value of u vc can be determined from these data and is used in Eq. (8.4) as indicated in Table 8.1, also reporting the value of L CG .

FTF measurements shown in Figs. 6.6 and 6.8 are reported in Fig. 8.4-left with respect to the Strouhal number St a . This scaling law is well suited for the congurations where the bulk ow velocity U b is modied, but it does not allow to reduce the data for the other two congurations, where the injector diameter D o or the swirler to injector outlet distance δ are modied.

Inspired from Kim and Santavicca, 2013a, a new denition of the Strouhal number is considered that takes into account the distance δ from the swirler exit to the injector outlet section. The main dierence with Eq. (8.2) is that the reference velocity is not U b , but two convective velocities are considered. Azimuthal perturbations inside the injector are convected at the phase velocity u c . Large vortical structures shed from the burner rim are convected at the velocity u vc through the combustion chamber : The velocity u c is determined from LES calculations and is shown in Fig. 7.13right. One can see that this velocity lies in between the average and the maximum velocity in the section where it is measured. As a consequence, a mean velocity u c = 1.2u is taken in Eq. (8.6), where u is the bulk ow velocity determined from mass balance in the injector cross section between the swirler exit and the combustion chamber backplane. Two dierent values of u c are used for the congurations of the injector featuring a dierent diameter over the section of length δ 1 and δ 2 (see Fig. Some dierences can also be identied in the FTFs in Fig. 8.4-right. This is probably due to the strong assumption made on the value taken for the convective velocity u c inside the injector. Some improvement would likely be obtained with a better knowledge of the value of u c . The transfer function measurements reported in chapter 6 showed that the FTF gain curve of swirling ames is characterized by peaks and valleys and that the frequencies corresponding to these values are shifted when the bulk ow velocity U b inside the injector or the distance δ between the swirler trailing edge and the combustion chamber backplane is modied. Two mechanisms are retained as dominant for the description of the observed ame dynamics: interferences between the acoustic pulsation and azimuthal ow disturbances generated at the swirler outlet and interaction of the ame with vortical structures generated at the burner rim.

St b = f ( δ u c + L CG u vc ). ( 8
In the literature, FTF measurements are usually scaled through a Strouhal number St taking into account a ame length scale and a characteristic convection velocity of the vortical structures shed from the burner rim and convected in the combustion chamber. Kim and Santavicca, 2013a also considered the response of the swirler. They introduced a new Strouhal number also considering the propagation time of vortical disturbances from the swirler to the combustor dump plane. They however considered the same characteristic velocity for the convection of vortical disturbances inside the injector and that of vortical structures in the combustion chamber.

These velocities can actually be quite dierent depending upon the injector geometry. Following the analysis of Kim and Santavicca, 2013a, a slightly different denition of the Strouhal number, based on the dierent convective time scales of the two dominant phenomena, is shown to be suited to nondimensionalize FTF results obtained for dierent values of the injection diameter D o , bulk ow velocity U b and distance δ between the swirler outlet and the injector outlet section.

Chapter 9

Impact of central blu body dierent ame responses are found to be related to the dynamics of the internal recirculation region, which response strongly diers depending upon the mechanism adopted to stabilize the ame.

Introduction

The frequency response of premixed swirling ames submitted to ow rate modulations is a topic of high scientic and technical interest due to the problems raised by combustion instabilities in gas turbines [START_REF] Lieuwen | Combustion Instabilities: Basic Con-References[END_REF][START_REF] Huang | Dynamics and stability of lean-premixed swirl-stabilized combustion[END_REF][START_REF] Poinsot | Prediction and control of combustion instabilities in real engines[END_REF]. This response is often characterized by a Flame Transfer Function (FTF) or more recently by a Flame Describing Function (FDF) when the level of ow disturbances is considered [START_REF] Candel | Dynamics of Swirling Flames[END_REF].

Changing the shape of the FTF/FDF by modifying the injector design is a way to augment the stability margins of a combustor. However, there is still no systematic way to make these changes, because the dynamics of swirling ames is not fully understood [START_REF] Thumuluru | Characterization of acoustically forced swirl ame dynamics[END_REF][START_REF] Candel | Dynamics of Swirling Flames[END_REF]. Improved combustor stability is thus gained by a costly trial and error iterative process and there is a need for better knowledge of the fundamental mechanisms controlling the shape of the FTF of swirling ames.

The FTF of premixed swirling ames can be determined analytically in simplied congurations [START_REF] Hirsch | Inuence of the swirler design on the ame transfer function of premixed ames[END_REF][START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF][START_REF] Acharya | Swirl effects on harmonically excited, premixed ame kinematics[END_REF] or by numerical ow simulations in more complex geometries [START_REF] Tay Wo | Identication of Flame Transfer Functions Using LES of Turbulent Reacting Flows[END_REF][START_REF] Biagioli | Dynamic response of turbulent low emission ames at different vortex breakdown conditions[END_REF][START_REF] Acharya | Eect of azimuthal ow uctuations on ow and ame dynamics of axisymmetric swirling ames[END_REF]. Most often this response is determined experimentally by using well proven optical techniques [START_REF] Kulsheimer | Combustion Dynamics of Turbulent Swirling Flames[END_REF]Kim et al., 2010a;[START_REF] Palies | Experimental study on the eect of swirler geometry and swirl number on ame describing functions[END_REF][START_REF] Osi¢ | Response of a swirl-stabilized ame to simultaneous perturbations in equivalence ratio and velocity at high oscillation amplitudes[END_REF] even in engine like conditions [START_REF] Schuermans | Thermoacoustic Modeling of a Gas Turbine Using Transfer Functions Measured Under Full Engine Pressure[END_REF].

Since shear layers are highly responsive to acoustic forcing, the FTF of premixed ames stabilized by a blu body is mainly controlled by the shedding of large coherent structures, which are then convected by the mean ow and roll-up the ame. This ame roll-up process around a coherent vortical structure constitutes the main contribution controlling the FTF phase lag of premixed laminar [START_REF] Durox | Combustion dynamics of inverted conical ames[END_REF] and turbulent non-swirling jet ames [START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF]. It also constitutes one of the fundamental process controlling the dynamics of premixed swirling ames (Palies et al., 2011e;[START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]Oberleithner et al., 2015).

It has been demonstrated that the response of the swirling vane needs to be taken into account in the dynamics of swirling ames [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Palies et al., 2010b;[START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. Vortical transverse perturbations triggered by the axial ow disturbances at the swirler outlet lead to oscillations of the swirl level at the burner outlet. This in turn leads to oscillations of the ame angle at the anchoring point location (Palies et al., 2011e). This swirl oscillation mechanism and its impact on the FTF have been identied in sev-CONTENTS 159 eral setups in which the ame is stabilized by a central blu body [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Palies et al., 2010b;[START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. The same dynamics is observed when the acoustic pulsation is introduced from the upstream or downstream side of the swirler [START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF].

In high power systems, the ame is most often fully aerodynamically stabilized without the help of any solid central insert. [START_REF] Giuliani | Inuence of pulsed entries on a spray generated by an airblast injection device: An experimental analysis on combustion instability processes in aeroengines[END_REF] also report large swirl number oscillations in the response of an aeronautical injector powered by kerosene when it is submitted to ow rate modulations. They however provide no FTF data. [START_REF] Biagioli | Dynamic response of turbulent low emission ames at different vortex breakdown conditions[END_REF] analyzed the FTF of aerodynamically swirl-stabilized ames and found that the position of the Internal Recirculation Zone (IRZ) and the ame leading edge respond to the acoustic forcing by a large axial motion, but the tangential ow component is not considered in their analysis and one cannot conclude about the role of swirl oscillations.

There is yet no detailed investigation on the impact of swirl number oscillations on the FTF of swirling ames aerodynamically stabilized away from all solid components. This response is analyzed here for ames stabilized either only by a blu body, only by swirl or by both swirl and blu body. The premixed ames investigated feature the same equivalence ratio and the same thermal power.

The experimental setup and diagnostics are presented in section 9.2, followed by a description in section 9.3 of their ame structure in the absence of forcing. Their frequency response is analyzed in section 9.4 for the dierent injectors tested. The ow and ame dynamics at selected frequencies are investigated in section 9.5 to infer the swirl number uctuations and the mechanisms controlling the response of these ames. Conclusions are nally drawn in section 9.6.

Congurations explored

The experimental setup is sketched in Fig. 9.1. The burner is powered by a methane/air mixture. Experiments are conducted at a xed equivalence ratio φ = 0.82 and bulk velocity U b = 5.44 m/s (T = 20 o C and p = 1 atm) in the D = 22 mm diameter section before the swirler unit. These conditions correspond to a constant thermal power P = 5.44 kW assuming total combustion. Two dierent radial swirlers can be xed in the injection unit. They both feature six radial injection channels of d c = 6 mm diameter. The design of swirlers SW 0 and SW 3 only dier by the distance x indicated in Fig. 9.1. In the rst device, designated as SW 0 , the channels are aligned with the radial direction (x = 0 mm). In the second one, designated as SW 2 , the channels are shifted from the radial direction by x = 6 mm, to impart a strong rotation to the ow. For ame S 3 -as at the bottom in Fig. 9.1, the central rod is removed and the ame is fully stabilized aerodynamically. The central injection tube is in this CONTENTS 161 case slightly modied and comprises a tube with diameter D = 22 mm over a rst section of length L 1 = 22 mm, followed by a nozzle of length L 2 = 34 mm, which is terminated by a diverging cup with an angle β = 15 • over an height of 6 mm. The nozzle throat diameter is in this case D 0 = 12 mm in Fig. 9.1.

The combustion chamber has an 82 mm square cross-section and a length of 150 mm, and is equipped with four quartz windows. At the base of the burner, a loudspeaker (Monacor SP-6/108PRO, 100 W RMS) is mounted to pulsate the ow. The velocity is measured with a hot wire anemometer probe (Dantec Dynamics -Probe 55P16 with a mini-CTA 54T30) below the swirler unit where the velocity has a top hat prole. A photomultiplier (Hamamatsu, H5784-04), equipped with a narrowband lter (Asahi Spectra, ZBPA310) centered around 310 nm and with a 10 nm bandwidth, is used to record the OH* chemiluminescence signal.

A 2D-2C Particle Image Velocimetry (PIV) system is also used to analyze the ow structure at the injector outlet under cold ow operation. Small oil droplets of diameter 3 µm are, in this case, seeded in the ow. The PIV system consists of 2×400 mJ Nd:YAG laser doubled at 532 nm operated at 10 Hz and a 2048×2048 px 2 CCD camera (Dantec Dynamics, FlowSense EO 4M). Two dierent optical setups are used for longitudinal and transverse measurements, with a time delay between the two laser pulses ∆t =10 µs and a pixel pitch of 27.9 px/mm in the rst case and ∆t =25 µs with a pixel pitch of 40.1 px/mm in the second one. Eight hundred images are taken to obtain converged mean and rms values of the velocity eld, which is deduced from the cross-correlation of the PIV images by a three passes window deformation technique (from 64×64 px 2 to 16×16 px 2 interrogation areas), with an uncertainty of 0.1 px on the calculated displacement.

An intensied CCD camera (Princeton Instruments, PI-MAX 4, 1024×1024 px 2 ), mounted with an UV objective (Nikkor 105 mm f/4.5) and equipped with the same lter as the photomultiplier, is also used to analyze the ame structure under steady and forced conditions. Phased averaged images of the OH* signals and the PIV elds are synchronized by the signal driving the loudspeaker.

Steady injection conditions

The PIV data gathered in the axial plane and a transverse plane 2 mm above the top cone of the central blu-body (ames S 0 -bb, S 3 -bb) and 2 mm above the injector outlet (ame S 3 -as) are rst used to determine the swirl number S [START_REF] Candel | Dynamics of Swirling Flames[END_REF] at the injector outlet : S=0.20 for S 0 -bb, S=0.80 for S 3 -bb and S=0.75 for S 3 -as, with a relative precision ±3%.

The swirl level for ame S 0 -bb slightly diers from zero due to small imperfections in the swirler manufacturing. Several PIV measurements were made to check this feature that was found to be reproducible, with the same velocity prole, from tests to tests by mounting and demounting the swirler and the rod. Nonetheless, the swirl number S = 0.2 remains in this case small and the conguration S 0 -bb will be referred in the following as a non-swirling ame.

Eects of the swirl number S on the shape taken by the ames are shown in Fig. 9.1. The ame S 0 -bb is anchored on the blu body at the top in Fig. 9.1 with a relatively narrow reaction layer spreading over a long distance in the wake of the central blu-body. The ame S 3 -bb produced by the same injector but with a higher swirl S = 0.8 is more compact in Fig. 9.1. When the central rod is removed, Fig. 9.1 shows that the lifted ame S 3 -as has about the same axial extent as ame S 0 -bb but with an emission intensity peaking in the central region.

Flame transfer functions

The FTF of the three preceding ames is determined from the velocity signal measured by the hot wire anemometer and the OH* chemiluminescence intensity I measured by the photomultiplier gathering light from the whole combustion region. This signal is assumed to be a good tracer of the heat release rate. Flames are excited by the loudspeaker in the frequency range 20-260 Hz with a constant modulation level u /u=0.3, where u and u denote the mean and root-mean-square (RMS) values of the velocity signal measured by the hot wire. The FTF is deduced from the cross-and power-spectral densities of the hot wire and photomultiplier signals evaluated at the forcing frequency f :

F(f ) = I /I u /u = G(f ) exp(iϕ(f )) (9.1)
Results are plotted in Fig. 9.2. The FTF gain curve of ame S 0 -bb shows a typical low-pass lter behavior with a large gain overshoot at low frequencies, a well known feature of the acoustic response of laminar and turbulent premixed non-swirling V-ames anchored on a central blu body [START_REF] Schuller | A unied model for the prediction of laminar ame transfer functions: comparisons between conical and V-ame dynamics[END_REF][START_REF] Durox | Combustion dynamics of inverted conical ames[END_REF][START_REF] Balachandran | Experimental investigation of the nonlinear response of turbulent premixed ames to imposed inlet velocity oscillations[END_REF].

The FTF of the ame S 3 -bb obtained with the same injector design, but a swirl level S = 0.80, diers from ame S 0 -bb by several aspects in Fig. 9.2.

The FTF gain of S 3 -bb shows a succession of a peak, a valley at f 0 = 96 Hz and another peak over a short frequency range, over which the non-swirling ame S 0 -bb response remains maximum. These features are now well understood and documented in a series of theoretical, numerical and experimental studies [START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]Palies et al., 2010b;[START_REF] Palies | Modeling of premixed swirling ames transfer functions[END_REF][START_REF] Kim | Interference mechanisms of acoustic / convective disturbances in a swirl-stabilized lean-premixed combustor[END_REF]. They result from interferences between axial and azimuthal velocity uctuations that are produced at the swirler outlet. It has been demonstrated that the frequency f 0 at which the FTF gain of a swirling ame reaches its minimum is obtained for the largest swirl number oscillation, i.e. a situation where azimuthal and axial velocity uctuations are out of phase at the burner outlet (Palies et al., 2010b).

The frequency f 0 can be roughly estimated by considering the phase-lag ϕ u θ -u z between azimuthal and axial velocity uctuations at the injector outlet (Palies et al., 2011a;[START_REF] Kim | Interference mechanisms of acoustic / convective disturbances in a swirl-stabilized lean-premixed combustor[END_REF]:

ϕ u θ -u z = 2πf 0 δ 1 u c - 1 c (9.2)
where u c is the convection velocity of azimuthal disturbances and c u c is the speed of sound. The condition leading to the highest swirl number oscillation amplitude, i.e. a minimum FTF gain, corresponds to ϕ u θ -u z = π. The distance δ in Eq. (9.2) is between the top of the swirler injection channels, where azimuthal disturbances are generated, and the combustor backplane. It is thus slightly smaller than the distance L = 56 mm between the swirler and the combustor backplanes. The velocity u c is close to the maximum velocity reached by the ow in the injection channel (Palies et al., 2010b;Palies et al., 2011a;[START_REF] Acharya | Eect of azimuthal ow uctuations on ow and ame dynamics of axisymmetric swirling ames[END_REF]. This velocity remains unknown in the present study, but a rough estimate is made here by taking it equal to the highest axial velocity u c = 9 m/s measured close to the injector outlet with PIV . The same choice was made in Palies et al., 2010b. One nds f 0 = 90 Hz for u c = 9 m/s and δ = 50 mm. This estimate is close to f 0 = 96 Hz determined experimentally in Fig. 9.2 for ame S 3 -bb. The largest uncertainty comes from the value of the convection velocity u c , which depends on the ow structure between the swirler injection channel outlets and the burner outlet.

The FTF of the swirling ame S 3 -as obtained without central insert does not feature any signicant FTF gain drop in Fig. 9.2 even though the swirl level S = 0.75 is close to the value S = 0.80 for ame S 3 -bb anchored on the blu body. One may again roughly estimate the frequency f 0 associated to the largest swirl number oscillations. Considering the reduction of the injection tube diameter, one now takes u c,1 = 5.44 m/s over δ 1 = 16 mm and u c,2 = 18.2 m/s over δ 2 = 34 mm deduced from mass balance (see Fig. 9.1) and nds f 0 = 104 Hz. This frequency is well inside the range investigated but, nonetheless, there is no drop of the FTF gain in Fig. 9.2 for ame S 3 -as. Other interesting features, which seem to have not been reported previously, are observed for the FTF phase lag in Fig. 9.2. The phase lag curve of the nonswirling S 0 -bb and highly swirled S 3 -bb ames anchored on the central blu body are the same at low frequencies f < f 0 . This is not an obvious feature since the time lag τ appearing in the FTF phase lag ϕ = 2πf τ is generally proportional to the ame length L f and to the inverse of the bulk ow velocity 1/U b . In Fig. 9.1, obtained for steady injection conditions, the ame length L f and ame aspect ratio largely dier between S 0 -bb and S 3 -bb while the bulk ow velocity remains the same. At higher frequencies f > f 0 , the FTF phase lag of the non-swirling ame S 0 -bb keeps increasing with an almost constant slope in the full frequency range investigated. An inection point and a rapid change of the phase lag characterize the response of the highly swirled ame S 3 -bb at frequencies close to f ∼ f 0 . At higher forcing frequencies, the data for the FTF phase lag of ame S 3 -bb are found to be parallel to the FTF phase lag plot of the non-swirling ame S 0 -bb oset by a constant value of ϕ 0 -1.3 rad. The FTF phase lag of the aerodynamically stabilized swirled ame S 3 -as regularly increases with a dierent slope than ame S 0 -bb and does not exhibit any inection point.

Flame dynamics

Phase averaged images of the OH* chemiluminescence conditioned by the harmonic excitation are examined in Fig. 9.3 to elucidate some of the previous observations. The forcing level u /u = 0.30 RMS is the same as in Fig. 9.2 showing the FTF results. The intensied CCD camera is synchronized with the signal driving the loudspeaker at the bottom of the burner. Images are taken for each conguration at the same phases separated by a constant interval of 30 o . The phase angles are indicated in Fig. 9.3 with respect to the hot-wire signal below the radial swirler and a phase shift arises between ames forced at 96 Hz and 170 Hz which correspond to the minimum and maximum gain of the FTF for ame S 3 -bb. An Abel deconvolution reveals the trace of the ame luminosity in an axial plane crossing the burner axis for ames S 0 -bb and S 3 -bb. This post-processing was not possible for ame S 3 -as due to the too high intensity values close to the symmetry axis (see Fig. 9.1). The same color scale is used for all images to better highlight both the ame motion and changes of the ame luminosity during the forcing cycle.

The rst sequence in Fig. 9.3 highlights the large motion undergone by the non-swirling ame S 0 -bb at the forcing frequency f = 96 Hz when the FTF gain is maximum in Fig. 9.2. Large roll-up of the ame tip is seen at 195 • and 255 • . The ame is stretched in the vertical direction during the forcing cycle with relatively minor changes of the OH* luminosity.

The second and third sequences in Fig. 9.3 show the responses of ame S 3 -bb, at the FTF gain minimum at f 0 = 96 Hz and at the FTF gain maximum at f = 170 Hz in Fig. 9.2. The motion undergone by the ame does not dier signicantly at these two forcing frequencies, but it is mainly changes of the ame luminosity that explain the large dierences observed for the FTF gain at f 0 = 96 Hz and f = 170 Hz in Fig. 9.2. At f 0 = 96 Hz, there is a relatively weak ame roll-up motion accompanied by weak changes of the ame luminosity over the forcing cycle in Fig. 9.3. At f = 170 Hz, the ame roll-up process is a bit further pronounced, but the OH* luminosity undergoes large changes during the forcing cycle explaining the high value taken by the FTF gain at this frequency in Fig. 9.2.

The last sequence in Fig. 9.3 shows the dynamics of the aerodynamically stabilized ame S 3 -as at f = 96 Hz corresponding to its peak FTF gain value in Fig. 9.2. The ame is rolled-up by vortex interaction (75 • -195 • ) and is stretched in the vertical direction, but also undergoes large changes of its luminosity as ame S 3 -bb at f = 170 Hz. The position of the leading edge of ame S 3 -as also exhibits a large vertical oscillation during the forcing cycle, as highlighted by the white crosses in this sequence, while ames S 0 -bb and S 3 -bb remain anchored on the blu body in Fig. 9.3.

Further analysis is made at the forcing frequency f 0 = 96 Hz by determining the average ame position for each image sequences. This prole is obtained by nding the maximum row-wise intensity of the pixel luminosity, weighted by the distance from the burner axis as expressed by: I(r, x)2πrdr. A threshold of 25% is selected to delineate the lower (cross symbol) and upper (diamond symbol) ame boundaries. For ame S 3 -as in the last sequence in Fig. 9.3, the ame contour is used in place of the average ame position for the analysis. This post-processing is used to deduce the ame height H corresponding to the vertical distance between the upper (diamond sign) and lower (plus sign) ame to normalize the results. The three ames exhibit a modulation of their height H during the forcing cycle, but the oscillation amplitude H is a bit lower for S 3 -bb compared to S 0 -bb and S 3 -as. The swirling ame S 3 -bb stabilized by the blu body also features a large oscillation of the ame angle α b at its base at f 0 = 96 Hz. These ame angle oscillations are not observed for the nonswirling ame S 0 -bb and could not be determined for the fully aerodynamically stabilized ame S 3 -as.

Figure 9.4 also shows the evolution of the swirl number S at the injector outlet, determined, for each selected phase in the cycle, by PIV measurements conducted in cold ow conditions, with a relative precision of ±5%. One clearly identies a large modulation of the swirl level at f = 96 Hz for the swirling ames S 3 -bb with and S 3 -as without blu body, while at the same frequency there are no swirl oscillations for the ame S 0 -bb. This analysis conrms that both ames, S 3 -bb and S 3 -as, undergo large swirl number oscillations at f = 96 Hz, but their FTF largely dier even though they share about the same swirl level S ∼ 0.8.

Finally, the axial velocity signal u z on the symmetry axis r = 0 is examined in Fig. 9.4. This signal measured 2 mm above the top cone is barely altered by the ow modulation at f 0 = 96 Hz for ames S 0 -bb and S 3 -bb anchored in the wake of the central blu body. This contrasts with the large oscillation in Fig. 9.4 observed for the same signal measured 2 mm above the injector outlet for the swirling ame S 3 -as without blu body. This large modulation is responsible for the displacement of the leading edge position of the aerodynamically stabilized ame S 3 -as in the bottom image sequences in Fig. 9.3.

Further analysis is now carried out under cold ow conditions by examining the dynamics of coherent vortical disturbances synchronized by the acoustic pulsation. Results are presented in Fig. 9.5. To identify vortical structures, the Q criterion [START_REF] Jeong | On the identication of a vortex[END_REF] is selected:

Q = 1 2 (|Ω| 2 -|S| 2 ) (9.3)
where S and Ω are the symmetric and anti-symmetric components of the velocity gradient respectively. Iso-contours of Q are inferred from PIV measurements in the axial plane. Only positive values of Q are retained. Negative values, in- dicating regions where shear is present but no swirling motion, are forced to a zero value.

In the rst, third and last sequences in Fig. 9.5, corresponding to FTF gain maxima in Fig. 9.2, large vortical structures generated at the rim of the injector are produced by the acoustic forcing and convected downstream in the chamber. In the second sequence in Fig. 9.5, corresponding to the FTF gain minimum in Fig. 9.2 for ame S 3 -bb at f 0 = 96 Hz, vortical structures are much weaker as emphasized by the much lower values taken by the Q criterion. As a consequence, the ame response remains low at this forcing frequency.

Isolines of axial velocity are also shown in Fig. 9.5, to highlight the dierent dynamics of the internal recirculation zone depending upon the ame stabilization mechanism. Flame S 0 -bb feature a small recirculation region in the wake of the blu-body. Flame S 3 -bb feature a larger IRZ, undergoing a apping motion during the forcing cycle, which is more evident at f 0 = 96 Hz. When the ame is stabilized without blu-body, S 3 -as, the IRZ is much thinner and oscillates vertically in and out of the injector.

These observations are confronted with current interpretations of the response of swirling ames associated with the combined eects of swirl number oscillations and ame vortex roll-up. Palies et al., 2011e explained that at the FTF gain minimum, large swirl number oscillations modulate the strength of the IRZ. This oscillation weakens the formation of eddies and also leads to ame base angle oscillations. When the ame angle uctuates, vortex growth is rapidly hindered by the ame apping motion and the ame response is low.

When the ame angle oscillation weakens, the response is high because vortices can fully develop before interacting with the ame. Interaction of vortex shedding and ame angle uctuations has been further analyzed in [START_REF] Bunce | Interaction Between Swirl Number Fluctuations and Vortex Shedding in a Single-Nozzle Turbulent Swirling Fully-Premixed Combustor[END_REF]. These authors argue that as the ame moves closer to the shear layer, the vorticity of the ow is dissipated due to its interaction with the ame and the ame response is low. In contrast, when the mean ame position is stationary, the vorticity of the ow is not dissipated before interacting with the ame, which leads to a large ame response.

The observations made in this work are for some aspects consistent with these conclusions, but also reveal new mechanisms. Figure 9.4 conrms that swirl number and ame base angle oscillations are large for ame S 3 -bb stabilized by a blu body at f 0 = 96 Hz where the FTF gain is at a minimum. It is however found that the formation of large vortical structures are damped even without combustion. This phenomenon is here not related to the apping motion of the ame, but is found to be related to the geometry of the injector. At the same excitation frequency, large vortical structures are shed from the injector without blu-body, while large swirl level oscillations are also observed. In this case, the response of ame S 3 -as remains high. This ame also exhibits a large vertical oscillation of its leading edge and a peak value of the FTF gain at f = 96 Hz. This analysis reveals that the mechanisms controlling the frequency response of swirled ames largely dier when they are stabilized by a blu body or when they are stabilized aerodynamically, the main dierences being related to the dynamics of the IRZ.

Conclusion

Transfer functions of ames stabilized with dierent injector designs have been investigated for dierent swirl levels, with and without a central insert in the injector. Depending on the stabilization mechanism, ame vortex roll-up, oscillations of the ame base angle induced by swirl level oscillations and vertical oscillations of the ame leading edge, are found to be the competing mechanisms controlling the ame response. When the FTF gain is at a maximum, the three ames investigated are strongly modulated by the interaction with large vortical structures convected in the external shear layer of the ow, regardless of changes of the swirl level and stabilization mechanisms. In contrast, the formation of large vortical structures is hindered, even in cold ow conditions, at the frequency corresponding to a minimum in the FTF gain curve, which is observed in the present study only for the swirling ame stabilized by a blu body. At this frequency, large swirl number oscillations lead to a modulation of the ame base angle and weak vortex formation. At the same frequency, but for the aerodynamically stabilized ame without central insert, swirl number oscillations of the same order of magnitude are observed, but the gain of the FTF remains high. In this case, large vortical structures are shed and the ame leading edge oscillates vertically. The origin of the low response of swirling ames at specic frequencies is found to not only be related to large oscillations of the swirl level, but also to the ame stabilization mechanism and more specically to the dynamics of the internal recirculation region.

Conclusion

The dynamics of lean premixed swirling ames was investigated in this manuscript.

The main concern was to assess the eects of dierent injection conditions on the properties of ame stabilization, thermoacoustic instability and ame response to acoustic perturbations. In particular we considered eects of dierent (i) design of the swirler device, (ii) dimensions of the injector, (iii) bulk ow velocities and (iv) ame stabilization mechanisms.

Flame stabilization

A stabilization chart was obtained and three ame regimes were identied depending on the value of the swirl number and a normalized injector cross section area:

• at low level of swirl (S < 0.5), ames are elongated in the downstream direction with a shape similar to that of non-swirling V-ames • in the intermediate swirl range (0.5 < S < 0.8) ames are well stabilized with a compact shape • when the swirl number is too elevated (S > 0.8) ames ashback inside the injector.

These limits on the swirl number should only be taken as an indication and are also dependent upon the main dimensions of the injector. For a xed swirl number, an injector with an higher nozzle cross section has more propensity to ashback.

Thermoacoustic instability

One of the major drawback for the application of lean premixed combustion systems, is their sensitivity to thermoacoustic instabilities. In this work, we concentrated on the eects of dierent geometries of the injector on this phenomenon. In particular, we showed that it was possible to lower the intensity of the instability by properly modifying the distance between the swirler device and the combustion chamber backplane. On the other hand, when the dimension of the injector are xed, a dierent design of the swirler did not lead to any important dierence on the properties of the thermoacoustic instability.

Flame response to acoustic perturbations FTF measurements were conducted by submitting ames to well controlled acoustic perturbations. It was shown that the FTF gain curve of highly swirled ames is characterized by peaks and valleys while, when the swirl number is low (S = 0.20), the FTF behaves like a low-pass lter, with the gain increasing to reach a maximum higher than unity and then constantly decreasing towards zero for increasing forcing frequencies. Surprisingly, the same low-pass lter behavior was observed for an high swirling ame fully aerodynamically stabilized, without the help of a blu-body. A low FTF gain value in the low frequency range could only be observed for highly swirled ames stabilized with the help of a blu-body. The FTF phase-lag is characterized by a linear behavior and by a jump with an inection point around the minimum gain frequency, if this minimum is present.

In terms of the eects of dierent injector geometries, it was observed that:

• as the swirl number is increased, either by an increase of the injector exit diameter, a decrease of the blu-body diameter or a dierent swirler device, the minimum gain value is decreased and the phase-lag jump increased • by modifying the velocity inside the injector, either by a change of injector diameter or bulk ow velocity, it is possible to shift the FTF curves in the frequency range • the same eect can be obtained by modifying the distance between the swirler and the combustion chamber backplane.

Mechanisms behind the acoustic response

Phase-locked (i) OH* chemiluminescence images, (ii) Particle Image Velocimetry (PIV) measurements and (iii) Large Eddy Simulations (LES) were exploited for an interpretation of the main mechanisms behind the acoustic response of lean premixed swirling ames. The focus was on the dierences between low and high value of the FTF gain, to assess those conditions possibly leading to a reduction of the sensitivity of LPM systems to acoustic waves.

Qualitatively, the motion of the investigated ames at these two peculiar frequencies is not dierent. The ame motion is the results of swirl number uctuations and interaction with vortical structures, leading to a modulation of ame length and angle. Swirl uctuations are due to the interaction of axial acoustic and azimuthal convective perturbations, which are formed at the swirler exit when this is impinged by acoustic waves. Due to the convective nature of azimuthal perturbations, this interaction depends upon swirler-combustion chamber distance and mean ow velocity inside the injector. This explains the eects on the FTF of these parameters.

Quantitatively, uctuations are stronger at frequencies corresponding to a maximum gain value. It was shown that this is due to a dierent strength of vortical structures at the two frequencies, which is not linked to ame/ow interaction since it was observed even in cold ow conditions. By post-processing of LES data, this was linked to a dierent phase-shift between the axial and the azimuthal velocity perturbations, that lead to a dierent shape of the velocity proles at the injector rim. The low strength of vortical structures at the FTF gain minimum frequency is behind the low ame response and the drift from linearity of the FTF phase-lag.

By considering the time scales of these two dominant mechanisms (interferences between the acoustic pulsation and the azimuthal ow disturbances generated at the swirler outlet and interaction of the ame with vortical structures generated at the burner rim), a nondimensionalization of the FTF results obtained with dierent injection conditions could be obtained.

The ame motion of a fully aerodynamically stabilized ame, analyzed at last, is somehow dierent. In this case, a vertical oscillation of the ame leading edge is observed as a consequence of the vertical oscillation of the internal recirculation region (IRZ) while, when the blu-body is present, this vertical oscillation is hindered and the IRZ mainly oscillates laterally.

Perspectives

• All the results presented in this manuscript refer to fully premixed system at a xed value of the equivalence ratio (Φ = 0.82). It would be interesting to see if the mechanisms observed can be conrmed for dierent values of Φ. Even more interesting would be to conduct a same analysis for technically premixed systems.

• FTF measurements are a powerful tool to analyze the response of the ame to incoming perturbations, but are time consuming. A step forward towards the identication of the main parameters controlling this response was taken in this manuscript, but work still need to be done. In particular, the dierent behavior of ames stabilized with or without the help of a blu-body, deserves further attention.

• The mechanisms leading to a dierent strength of vortical structures at the FTF gain minimum and maximum frequencies were identied through PIV and LES in cold ow conditions. It would be interesting to check whether the same conclusions could be drawn when performing the same analysis in reactive conditions.

• FTF measurements were performed only in the low and intermediate frequency range in this work. It would be interesting to extend these data to the high frequency range and see if new minimum or maximum FTF gain values are present. In this case, it would be worth to conduct the same analysis at these other peculiar frequencies and check if the same mechanisms are in action.

• It would be interesting to compare LES with experimental data for the conguration without the blu-body, in cold but also reactive conditions. This analysis would show the dierent ow properties inside the injector, leading to the dierent responses of swirling ames stabilized with/without blu-body.

instabilities. This was checked for two conguration with dierent exhaust tube length.

This check was important to conrm that the analysis of ame motion conducted by externally exciting the ow with a loudspeaker, correctly reproduces the same mechanisms observed during self-triggered instabilities.

Figure 1 :

 1 Figure 1: World total primary energy supply by fuel (Mtoe) from 1971 to 2015. Adapted from IEA, 2017a.

Figure 2 :

 2 Figure 2:

Figure 3 :

 3 Figure 3: Mean temperature elds and streamline patterns for two dierent swirl numbers. Reproduced from Huang and Yang, 2005.

Figure 4 :

 4 Figure 4: Schematic diagram of processes leading to the formation of an IRZ: large tangential velocities create a centrifugal pressure gradient and sub-atmospheric pressure near the central axis; axial decay of tangential velocity causes decay of radial distribution of centrifugal pressure gradient in axial direction; thus, an axial pressure gradient is set up in the central region towards the swirl burner, causing reverse ow. Reproduced from Syred, 2006.

Figure 6 :

 6 Figure 6: Flame describing function measurements for a non-swirling (left, Durox et al., 2005) and a swirling ame (right, Palies et al., 2010a) anchored on a central blu body.

Figure 7 :

 7 Figure 7: Schematic of the swirl-stabilized combustor used in the analysis of Bellows et al., 2007.

Figure 8 :

 8 Figure 8: Dependence of CH* oscillation amplitude and length, L, of the ame front, and u'-CH*' phase angle upon velocity oscillation amplitude. Left: f = 130 Hz. Right: f = 410 Hz. Reproduced from Bellows et al., 2007.

Figure 9 :

 9 Figure 9: Instantaneous OH PLIF images showing evolution of ame dynamics over one cycle of acoustic forcing for (a) low and (b) high velocity oscillation amplitudes. Top: f = 130 Hz. Bottom: f = 410 Hz. Reproduced from Bellows et al., 2007.

Figure 10 :

 10 Figure 10: Test rig used at TU Berlin with measurement devices. Reproduced from Schimek et al., 2011.

Figure 11 :

 11 Figure 11: (a) Amplitude of the ame describing function (FDF) of the perfectly premixed swirl-stabilized ame. (b) Phase delay between the inlet perturbation and the global heat release rate uctuation. Reproduced from Oberleithner et al., 2015.

Figure 12 :

 12 Figure 12: Color-coded contours of the phase-averaged OH* images of the ame forced at 254 Hz. The contour lines represent an isocontour of the Q-criterion derived from phase-averaged PIV snapshots. The rst row shows the root mean square of the OH* uctuations indicating regions of highest heat release rate uctuations. Reproduced from Oberleithner et al., 2015.

Figure 13 :

 13 Figure 13: Block diagram representation of mechanisms generating heat release rate uctuations in swirling ows. Reproduced from Candel et al., 2014.

Figure 14 :

 14 Figure14: Sketch of the burner used for the analysis of[START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF]

Figure 15 :

 15 Figure 15: FTFs for three dierent positions of the axial swirl generator. Reproduced from Komarek and Polifke, 2010.

Figure 16 :

 16 Figure 16: (a) Schematic view of the burner and diagnostics used for measurements made by Palies et al., 2010b. (b) Close-up view of the upper part of the burner. All size are in millimeters. Reproduced from Palies et al., 2010b.

  The burner was fed by a methane and air mixture at a xed equivalence ratio of φ = 0.70. Two operating conditions with two bulk velocities of U b = 2.67 m/s and U b = 4.13 m/s were considered. The acoustic forcing was provided by a loudspeaker mounted at the bottom of the burner. The velocity uctuation was measured with an hot wire probe below the swirling vane and the heat release rate oscillations with a photomultiplier equipped with an OH* lter. The FDF measurements, performed for the two operating conditions they explored, are reported in Fig.17. Both FDFs are characterized by maximum and minimum FDF gain values. As for the results of[START_REF] Komarek | Impact of Swirl Fluctuations on the Flame Response of a Perfectly Premixed Swirl Burner[END_REF], by increasing the time required for the convective transport of a uctuation in swirl number to the ame, the local minimum of the FTF gain moved to lower frequencies. InPalies et al., 2010b, this was obtained by reducing the bulk ow velocity. The non-linear behavior of the ame response was also observed for both operating

Figure 17 :

 17 Figure 17: Flame describing function for two dierent bulk ow velocities. Reproduced from Palies et al., 2010b.

Figure 18 :

 18 Figure 18: View of the two interrogation windows superimposed on ame A at f = 60 Hz. UW and LW respectively designate the upper and lower windows. Reproduced from Palies et al., 2010b.

Figure 19 :

 19 Figure 19: Heat release rate signals reconstructed from the light emission detected in the upper and lower windows. Reproduced from Palies et al., 2010b.

Figure 20 :

 20 Figure 20: Cross section of the nozzle and combustion chamber. Reproduced from Bunce et al., 2013.

Figure 21 :

 21 Figure 21: Flame transfer function gain and phase versus the forcing frequency. Reproduced from Bunce et al., 2013.

Figure 22 :

 22 Figure 22: Envelope of the mean ame position over one period of forcing at (a) the rst gain minimum, (b) the gain maximum and (c) the second gain minimum. Reproduced from Bunce et al., 2013.

Introduction CH 4 Figure 23 :

 423 Figure23: Experimental setup used by[START_REF] Gaudron | FLAME DE-SCRIBING FUNCTIONS OF A CONFINED PREMIXED SWIRLED COMBUSTOR WITH UPSTREAM AND DOWNSTREAM FORCING[END_REF]

Figure 25 :

 25 Figure 25: Experimental setup used for the study of fully aerodinamically stabilized swirling ames in Durox et al., 2013.

Figure 26 :

 26 Figure 26: Long exposure time ame images as a function of the swirler blade angle. Reproduced from Durox et al., 2013.

Figure 27 :

 27 Figure 27: Flame Transfer Function for dierent swirler blade angles. Reproduced from Durox et al., 2013.

Figure 1 . 1 :

 11 Figure 1.1: Burner sketch with alimentation system.

Figure 1 . 2 :Figure 1 . 3 :

 1213 Figure 1.2: Picture of the NoiseDyn burner (left) and corresponding view obtained with the CAD software Inventor (right).

Figure 1 . 4 :

 14 Figure 1.4: Swirler sample geometry and parameters of the dierent swirlers tested.

Fig. 1 .

 1 4. The dimensions of the external (D e = 22 mm) and internal (D i = 6 mm) diameter of the annular channel shown in Fig. 1.4, are xed for all the swirler designs. As shown in Fig. 1.3, a central insert of diameter D i = 6 mm ending with a top cone of diameter C, is generally xed to the swirler. For the analysis of aerodynamically stabilized ames without the help of a central insert, reported in chapter 9, the central hole of diameter D i = 6 mm is closed with a screw. The rate of rotation imparted to the ow is characterized by the swirl number S, comparing the axial ux of tangential momentum G θ to the axial momentum ux G z , S = 2G θ /(D o G z ), where D o is the outer diameter of the injector, as shown in Fig. 1.3.

Figure 1 . 5 :

 15 Figure 1.5: Sketch showing the velocities used for the swirl number estimation.

4 )

 4 Analytical estimates obtained with this assumptions are shown in Fig.1.6 for swirler SW 1 and SW 3 as a function of the injector exit radius R o . As expected, S increases for increasing values of H leading to higher azimuthal velocities and for increasing values of R o leading to smaller axial velocities. The swirl number was also determined by Laser Doppler Velocimetry (LDV) or Particle Image Velocimetry (PIV) measurements for the main congurations of the injector investigated in this work. Details about these measurements are given in chapter 2, but it is already interesting to compare them with the analytical estimates just introduced.LDV measurements of the axial and azimuthal velocity components are presented in Fig.1.7 for swirlers SW 1 and SW 3 , in an injector with an outlet radius R o = 10 mm and a central insert terminated by a cone of C = 12 mm diameter. The measurements are conducted in cold ow conditions at 1.5 mm above the top cone of the central blu-body, corresponding to a distance of 4.0 mm from the combustion chamber backplane (Fig.1.3). The corresponding bulk ow velocity at the hot wire location is U b = 5.44 m/s. The axial velocity components are almost superposed for the two swirlers SW 1 and SW 3 in the left in Fig.1.7. In these two cases, the swirler designs are too close (see Fig. 1.4) to obtain a major modication of the axial ow component. On the other hand, in the right plot in Fig. 1.7

Figure 1 . 6 :

 16 Figure 1.6: Swirl number analytical estimates (Eq.(1.4)) for swirlers SW 1 (white circles) and SW 3 (black diamonds). Solid lines: estimates obtained with a uniform axial velocity prole and a linear azimuthal velocity prole. Dashed dotted lines: estimates with linear axial and azimuthal velocity proles (Eq.(1.5)). Superposed as blue (SW 1 ) and red (SW 3 ) crosses, the swirl numbers measured by LDV.

Figure 1 . 7 :Figure 1 . 8 :

 1718 Figure 1.7: Axial (left) and azimuthal (right) velocity components measured by LDV 4.0 mm above the injector outlet for swirlers SW 1 (white circles) and SW 3 (black diamonds), with an injector of R o = 10 mm and C = 12 mm and for a bulk ow velocity U b = 5.44 m/s.

4 .

 4 1.3), while the other was exposed to the atmosphere. Results are shown in Fig.1.8. These measurements were conducted in cold ow conditions by injecting increasing air mass ow rates in the burner. The bulk velocity U b indicated in Fig.1.8 is evaluated at the convergent outlet of 22 mm diameter from the air mass ow rate by assuming an atmospheric pressure value p atm = 101.325 kPa and a temperature T = 293 K. As expected, the pressure loss increases with the bulk velocity and is lowest for swirler SW 0 , featuring injection channels aligned with the radial direction of the reactant injection tube (H = 0, Fig.1.4). The pressure drop ∆P is the highest for SW 4 featuring injection channels the farthest apart from each other (H = 7 mm) and the lowest ow passage area A = nπD 2 s /The maximum pressure loss for SW 4 almost reaches 1200 Pa in Fig.1.8. For the experiments conducted in this work, the bulk velocity was always equal or lower than U b = 5.44 m/s, giving a negligible ∆P always lower than 400 Pa. In this respect, one can neglect eects of the pressure drop through the swirler, since ∆p/p < 1% in this study.

Figure 1

 1 Figure 1.9: Details of the injectors used to stabilize ames with the help of a blu-body (a) or fully aerodynamically (b).

Figure 1 .

 1 Figure 1.10: Pictures and geometrical details of the replaceable components of the injection unit. Dimensions in mm.

Figure 2 . 1 :

 21 Figure 2.1: Schematic of the main components used for ame transfer functions measurements and for ame imaging.

Figure 2 .

 2 2 shows two samples over 5 forcing periods of the velocity signals measured by the hot wire at f = 120 Hz and f = 200 Hz for a forcing level u /u = 0.30 RMS. The power spectral density shown at the bottom in Fig.2.2 indicate that the velocity forcing signal is harmonic, with a small harmonic content of at least two orders of magnitude lower than the forcing level at the forcing frequency. The velocity on the left and the right in the top

Figure 2 . 2 :

 22 Figure 2.2: Top: Examples of signals measured by the HW (black) and the PM (gray) for forcing frequencies f = 120 Hz (left) and f = 200 Hz (right) and for a forcing level u /u = 0.30 RMS. Bottom: corresponding power spectral densities.

Figure 2 . 3 :

 23 Figure 2.3: Example of Flame Transfer Function result from the signals shown in Fig. 2.2. The forcing level is u /u = 0.30 RMS.

Figure 2 . 4 :

 24 Figure 2.4: Schematic of the setup used for the LDV measurements. (a) Lateral view showing the λ = 514.5 nm laser component used to measure the axial velocity. (b) Top view showing the λ = 488 nm laser component used to measure the azimuthal and radial velocities.

Figure 2 . 5 :

 25 Figure 2.5: Mean u and root mean square u rms values of the axial, azimuthal and radial velocity components for steady operating conditions, measured by LDV at a distance of 1.5 mm from the top cone of the central rod. Measurements refer to a geometrical conguration with SW 3 , D o = 20 mm and C = 10 mm and to a bulk ow velocity U b = 5.44 m/s.

Figure 2

 2 Figure 2.6: PIV setup

Figure 2 . 7 :

 27 Figure 2.7: Examples of Mie scattering images obtained under non-reactive conditions using oil seeding in an axial (left) or a transverse plane (right).

Figure 2 . 8 :

 28 Figure 2.8: Examples of Mie scattering images obtained under reactive conditions using ZrO 2 as seeding in an axial plane.

Figure 2

 2 Figure 2.9: Steady state scan of the mean (top) and root mean square RMS (bottom) axial velocity for swirler SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s obtained from post-processing of 800 PIV images. Left: cold ow conditions. Right: reactive conditions (φ = 0.82).

Figure 2 .

 2 Figure 2.10: Grid used to obtain ame dimensions.

Figure 2 .

 2 Figure 2.11: OH* chemiluminescence for SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s, φ = 0.82. Top row: without acoustic forcing. Bottom row: f = 185 Hz, u /ū = 0.72 RMS. On the right an Abel deconvolution is applied to the line of sight integrated image shown on the left.

Figure 2 .

 2 Figure 2.12: Abel deconvoluted OH* chemiluminescence for SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s, φ = 0.82. The ame is forced with a harmonic signal at f = 185 Hz and u /ū = 0.72 RMS. The phase angle indicated on the gures is referred to the velocity signal recorded by the hot wire as shown in the bottom gure.

Figure 3 . 1 :

 31 Figure 3.1: Turbulence energy spectrum plotted as a function of wave numbers (proportional to the inverse of the length scales). RANS, LES and DNS are summarized in terms of spatial frequency range. k c is the cut-o wave number used in LES (log-log diagram). Reproduced from Poinsot and Veynante, 2005.

Figure 3 . 2 :

 32 Figure 3.2: Left: sketch of the burner geometry. Highlighted in blue, the region of the ow selected for the LES. Right: uid volume generated with Inventor corresponding to the area highlighted in blue on the left.

  3.1. It will be shown in Chapter 7 that these selections let us match the experimental data in terms of CONTENTS 63 1.5 mm

Figure 3 . 3 :

 33 Figure 3.3: Left: axial cut of the simulated geometry showing the mesh. Right: zoom showing the rened zones over the injector section.

Figure 3 . 4 :

 34 Figure 3.4: Cold ow, steady state comparison of the axial (top), azimuthal (center), and radial (bottom) velocity proles obtained with LDV measurements and LES at a distance of 1.5 mm from the blu-body for the conguration of interest: SW 3 , D o = 20 mm, C = 10 mm, δ = 50 mm. Left: mean values. Right: RMS values.

Figure 3 . 6 :Figure 3 . 7 :

 3637 Figure 3.6: Cold ow, steady state comparison of the azimuthal and radial velocity proles obtained with PIV measurements and LES at 2 mm from the blu-body. Conguration studied: SW 3 , D o = 20 mm, C = 10 mm, δ = 50 mm.

Figure 4 .

 4 Figure 4.1 shows, as an example, the LDV proles obtained for three of the dierent congurations analyzed. LDV measurements are conducted, in cold ow conditions, 1.5 mm above the top cone of the central blu-body (see sec. 2.2 in Chapter 2). The mean axial velocity proles u z , Fig. 4.1-left, feature a central recirculation zone with slightly negative velocities, while the mean azimuthal velocities u θ , Fig. 4.1-right, have a Rankine like shape prole in the center part of the ow.We rst consider the case when the swirler design SW is modied, but the cross-section area at the exit of the injector A 2 is xed. The mean axial velocity u z prole does not change, while the peak of the mean azimuthal component u θ augments from SW 1 to SW 2 and shifts to a slightly higher radial position. This is due to the increase of the distance 2H between the axes of the tangential inlets of the swirling vane. This leads to an increase of the swirl number S for SW 2 compared to SW 1 . We now examine the case when the swirler design SW is xed, but the cross-section area at the exit of the injector A 2

Figure 4 . 1 :

 41 Figure 4.1: Mean axial ūz and azimuthal ūθ velocity proles deduced from LDV for three selected congurations: SW 1 , D o = 20 mm, C = 12 mm (blue), SW 2 , D o = 20 mm, C = 12 mm (red), SW 2 , D o = 18 mm, C = 12 mm (green). The distance δ = 50 mm and the bulk ow velocity U b = 5.44 m/s at the hot-wire location, are xed.

Figure 4 . 2 :

 42 Figure 4.2: Swirl number eect on ame stabilization. (a) SW 1 , D o = 18 mm, C = 14 mm, S=0.48*, elongated ame close to blow o. (b) SW 3 , D o = 20 mm, C = 12 mm, S=0.73, well-stabilized ame. (c) SW 3 , D o = 22 mm, C = 10 mm, S=0.86*, ash back. The equivalence ratio φ = 0.82, the bulk ow velocity U b = 5.44 m/s at the hot-wire location and the distance δ = 50 mm, are xed.

Figure 4 . 3 :

 43 Figure 4.3: Stabilization chart plotting dierent ame regimes as a function of the normalized ow passage area ratio A 1 /A 2 and the swirl number S. The equivalence ratio φ = 0.82, the bulk ow velocity U b = 5.44 m/s at the hot-wire location and the distance δ = 50 mm are xed.

Figure 4 . 4 :Figure 4 . 5 :

 4445 Figure 4.4: Cold ow velocity eld in an axial (left) and a transverse (right) plane obtained from PIV measurements for four dierent injectors. First row: SW 0 , D o = 22 mm, C = 14 mm, S = 0.20. Second row: SW 3 , D o = 22 mm, C = 14 mm, S = 0.80. Third row: SW 3 , D o = 20 mm, C = 10 mm, S = 0.80. Fourth row: SW 3 , D o = 18 mm, C = 12 mm, S = 0.60.

  .5) and the swirl number S decreases as a consequence. The central recirculation region still spans the full axial scan for the conguration with swirler SW 3 , D o = 18 mm, C = 12 mm and a swirl number S = 0.60 shown in the last line in Fig. 4.4, but it is thinner in the radial direction compared to the results shown in the third line for the conguration with swirler SW 3 , D o = 20 mm, C = 10 mm and with S = 0.80.

D 0 =S

 0 22 mm, C = 14 mm, = 50 mm, U b = 5.44 m/s

Figure 4 . 7 :

 47 Figure 4.7: OH* ame light distribution for dierent congurations tested with different swirl numbers. First line: eect of nozzle diameter D o . Second line: eect of top cone central blu body diameter C. Third line: eect of swirling vane SW .

Figure 4 . 8 :

 48 Figure 4.8: OH* ame light distribution for dierent congurations tested. First line: eect of the distance δ between swirler exit and combustor backplane. Second line: eect of the injection velocity U b . The swirl number is xed in each line. The colorscale is the same as the one used in Fig. 4.7.

Figure 5 . 1 :Figure 5 . 2 :

 5152 Figure 5.1: Sketch of the burner used for combustion instabilities characterization.

Figure 5 . 3 :

 53 Figure 5.3: Abel deconvoluted OH* chemiluminescence for the ame stabilized with the injector conguration featuring SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s, φ = 0.82 without any exhaust tube (E = 0 mm in Fig. 5.1).

5 Figure 5 . 4 :

 554 Figure 5.4: Sound pressure spectrum measured in the combustion chamber after the thermalization time for dierent length E of the exhaust tube downstream the combustion chamber. Left: 1E. stable system. Right: 2E, 3E. Unstable systems.

Figure 5 . 5 :Figure 5 . 6 :Figure 5 . 7 :Figure 5 . 8 :

 55565758 Figure 5.5: Phase averaged OH* chemiluminescence of the ame during one cycle of oscillation for the ame stabilized with the injector conguration featuring SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s, φ = 0.82. Top: 2E = 440 mm exhaust length. Bottom: 3E = 660 mm exhaust length. The colorscale is the same as the one used in Fig. 5.3. The phase angle indicated on the gures is referred to the velocity signal recorded by the hot wire.

Figure 5 .Figure 5

 55 Figure 5.8 shows that the pressure uctuations measured by MC in the combustion chamber and ME in the exhaust tube are much higher than those measured by MB at the burner bottom and MHW at the hot wire location. Fluctuations as high as 1 kPa are measured in the rst minutes of the experiments, while they drop to about half that level at 500 Pa when t ≥ 20 minutes. The signals recorded by MB and MHW are almost out of phase with respect to the signals measured by MC and ME in Fig.5.8. Considering that the system is closed at the bottom by a rigid plate and open at its exhaust, the acoustic boundary conditions correspond to a closed-open system. More information on the acoustic characterization of this setup with detailed measurements of these acoustic boundary can be found in[START_REF] Merk | Measurement and Simulation of Combustion Noise and Dynamics of a Conned Swirl Flame[END_REF]. The modal structure depicted in Fig.5.8 sugests that this instability is coupled to the 3/4 wave mode of the system, with a pressure node close to the hot wire location. This has been conrmed by a set of other experiments in the work of Gaudron[START_REF] Gaudron | Acoustic Response of Premixed Flames Submitted to Harmonic Sound Waves[END_REF].

Figure 5 .

 5 Figure 5.10: OH* chemiluminescence for the ame stabilized with the injector conguration featuring SW 3 , D o = 20 mm, C = 10 mm, U b = 5.44 m/s, φ = 0.82 without any exhaust tube (E = 0 mm in Fig. 5.1). Left: δ = 35 mm. Right: δ = 50 mm. The colorscale is the same as the one used in Fig. 5.3.

Figure 5 .Figure 5 .

 55 Figure 5.11: Sound pressure spectrum measured in the combustion chamber (MC) at t = 40 minutes for three lengths 1E, 2E and 3E of the exhaust tube downstream the combustion chamber and for two values of the distance δ between the swirler and the chamber backplane. Left: δ = 35 mm. Right: δ = 50 mm.

Figure 5 .

 5 Figure 5.13: Sound pressure spectrum measured in the combustion chamber (left) at t = 40 minutes and temporal evolution of the frequency of instability and of the SPL in the combustion chamber (right) for two dierent swirler designs.

Figure 5 .

 5 Figure 5.14: OH* chemiluminescence for the ame stabilized with the injector conguration featuring SW 3 (left) or SW 1 (right), D o = 22 mm, C = 14 mm, U b = 5.44 m/s, φ = 0.82 without any exhaust tube (E = 0 mm in Fig. 5.1). The colorscale is the same as the one used in Fig. 5.3.

  ) the main dimensions of the radial swirler, (3) the diameter of the injection nozzle and (4) the design of the top cone of variable diameter of the central blu body. Experiments start by examining eect of the forcing level. All experiments are conducted for xed ow injection conditions with a bulk ow velocity U b = 5.44 m/s at the outlet of the convergent nozzle, except in a specic set of experiments conducted to analyze the eects of a change of this velocity.

Figure 6 . 1 :

 61 Figure 6.1: Experimental setup used to determine the FTF/FDF.

Figure 6 . 2 :Figure 6 . 3 :

 6263 Figure 6.2: OH* ame light distribution for the geometrical conguration featuring SW 3 , D o = 20 mm, C = 10 mm, δ = 50 mm. The bulk ow velocity is U b = 5.44 m/s and the equivalence ratio is φ = 0.82.

Figure 6 . 4 :Figure 6 . 5 :Figure 6 . 6 :

 646566 Figure 6.4: OH* ame light distribution for the geometrical conguration featuring SW 3 , D o = 20 mm, C = 10 mm, δ = 35 -50 mm. The bulk ow velocity is U b = 5.44 m/s and the equivalence ratio is φ = 0.82. The colorscale is the same as the one used in Fig. 6.2.

102 Chapter 6 -Figure 6 . 7 :

 667 Figure 6.7: Evolution of the frequency corresponding to the minimum FTF gain value for dierent values of the distance δ between the swirler exit and the combustion chamber backplane (left) and of the bulk ow velocity U b (right).

6. 4 . 1 Figure 6 . 8 :

 4168 Figure 6.8 presents the eects of modications of the injector exit diameter(D o = 18, 20, 22 mm) on the ame transfer functions for ames stabilized with SW 2 (left) or SW 1 (right), with a blu-body featuring a conical end piece diameter of C = 12 mm. The shapes of these ames for the congurations featuring the swirling vane SW 2 are reported in Fig.6.9, showing that as D o decreases and thus the swirl number decreases, the ame becomes more elongated, moves

Figure 6 .

 6 Figure 6.9: OH* ame light distribution for the geometrical conguration featuring SW 2 , D o = 18 -22 mm, C = 12 mm, δ = 50 mm. The bulk ow velocity is U b = 5.44 m/s and the equivalence ratio is φ = 0.82. The colorscale is the same as the one used in Fig. 6.2.

Figure 6 .

 6 Figure 6.10: Flame Transfer Function measurements. Geometrical congurations: SW 2 , D o = 20 mm, C = 10 -14 mm. Forcing level: u /u = 0.10 RMS.

Figure 6 .

 6 Figure 6.11: OH* ame light distribution for the geometrical conguration featuring SW 2 , D o = 20 mm, C = 10 -14 mm, δ = 50 mm. The bulk ow velocity is U b = 5.44 m/s and the equivalence ratio is φ = 0.82. The colorscale is the same as the one used in Fig. 6.2.

Figure 6 .Figure 6 .

 66 Figure 6.12: Flame Transfer Function measurements. Geometrical congurations: SW 0,1,3,4 , D o = 22 mm, C = 14 mm. Forcing level: u /u = 0.30 RMS.

Chapter 7 Flow

 7 and ame dynamicsIn this chapter we focus on the mechanisms leading to the high or low FTF gain values observed in chapter 6. Strong swirl number uctuations leading to a strong ame-ow interaction at the ame base and hindering the development of vortical structures generated at the burner rim are currently interpreted as the reason for the low FTF gain at peculiar frequencies. These uctuations are weaker at the FTF gain maximum frequency which means that vortical structures are free to fully develop and interact with the ame at its tip. These mechanisms are here questioned through a set of experimental and numerical observations. Phase-locked OH* chemiluminescence images rst reveal that the motion of highly swirled ames is qualitatively similar at the two frequencies of interest, while a general decrease of ame luminosity and ame volume uctuations is observed as the gain level is reduced. Cold ow, phase-locked PIV measurements in an axial and in a transverse plane are then considered. These measurements show a general reduction of the strength of vortical structures at the FTF gain minimum frequency with respect to the FTF gain maximum frequency as reported in the literature. Phase conditioned Large Eddy Simulations are lastly exploited in cold ow conditions at the two frequencies. These data rst conrm the PIV results in terms of dierent strength of vortical structures shed from the burner rim. More importantly, swirl number uctuations are found to be comparable at the two frequencies. This result is in contrast with the current understanding from the literature.

1 Figure 7 . 1 :

 171 Figure 7.1: Flame Transfer Function measurements. Geometrical congurations: SW 0 , D o = 22 mm, C = 14 mm, (blue); SW 3 , D o = 18 mm, C = 12 mm, (orange); SW 3 , D o = 22 mm, C = 14 mm, (red); SW 3 , D o = 20 mm, C = 10 mm, (green). The forcing level is xed to u /ū = 0.30 RMS.

Figure 7 . 2 :

 72 Figure 7.2: (a): Abel deconvoluted phase conditioned images of the ame OH * signal for a forcing level u /ū = 0.30 RMS. The phase angle is referred to the hot wire signal. The ame contours used to determine the ame volume V are superimposed on the ame images. (b): comparison between unforced ame (left of the chamber axis) and mean ame determined from the phase conditioned images (right of the chamber axis).

7. 2 . 2

 22 Dierence between high swirling ame dynamics -Eect of the injector unit geometry This behavior is conrmed in the rst and second rows in Fig.7.3 showing the response of a ame stabilized with swirler SW 3 , but an injector equipped with a nozzle cone diameter D o = 20 mm and a top cone diameter C = 10 mm. This ame features the same swirl level S = 0.80 as the previous one, but a dierent level of the minimum FTF gain in Fig.7.1. The ame luminosity along the ame arms again barely changes for the ow forced at f 0 = 110 Hz in the rst row in Fig.7.3. This low response explains the minimum FTF gain value reached at this frequency in Fig.7.1. This again contrasts with the large uctuations of the OH* intensity signal observed in the second row in Fig.7.3 when the ame is forced at f = 190 Hz that corresponds to a large FTF gain in Fig.7.1. The qualitative dierences between the motion executed by the ame at these two forcing frequencies f = 110 and 190 Hz are less pronounced.A last case is considered in the third and fourth sequences in Fig.7.3 for an injector still equipped with swirler SW 3 but with a slightly smaller nozzle diameter D o = 18 mm and a slightly larger top cone diameter C = 12 mm. This ame features a lower swirl level S = 0.60 than the ame shown at the top in Fig.7.3. In this latter case, both the ame motion and the OH* luminosity uctuations are important at the two forcing frequencies f = 110 Hz and f = 180 Hz investigated. Indeed, the FTF gain level remains high in Fig.7.1 at these two frequencies for this injector characterized by a moderately high swirl number S = 0.60.One can get further insight by a closer examination of the OH* light distributions averaged over the forcing excitation cycle that can be compared to the OH* light distribution in the absence of acoustic forcing. These images are gathered in Figs.7.2-b and 7.3-b.

  .80, D 0 =20 mm, C=10 mm SW 3 , S=0.60, D 0 =18 mm, C=12 mm

Figure 7 . 3 :

 73 Figure 7.3: (a): Abel deconvoluted phase conditioned images of the ame OH * signal for a forcing level u /ū = 0.30 RMS. The phase angle is referred to the hot wire signal. The ame contours used to determine the ame volume V are superimposed on the ame images. (b): comparison between unforced ame (left of the chamber axis) and mean ame determined from the phase conditioned images (right of the chamber axis).

Figure 7 . 4 :

 74 Figure 7.4: Evolution of the swirl number S, the ame height normalized uctuation H f /H f and ame angle normalized uctuation α /α at the ame base. First line: SW 0 , S = 0.20, D o = 22 mm, C = 14 mm, f = 96 Hz. Second line: SW 3 , S = 0.80, D o = 22 mm, C = 14 mm. Left: f = 96 Hz, right: f = 170 Hz. Third line: SW 3 , S = 0.80, D o = 20 mm, C = 10 mm. Left: f = 110 Hz, right: f = 190 Hz. Fourth line: SW 3 , S = 0.60, D o = 18 mm, C = 12 mm. Left: f = 110 Hz, right: f = 180 Hz. The phase is referred to the hot wire signal. Two cycles are represented for better readability.

Figure 7 . 5 :

 75 Figure 7.5: Evolution of the spatially-averaged volumetric heat release rate uctuation < q > / < q >, ame volume uctuation V /V and sum of these contributions. First line: SW 0 , S = 0.20, D o = 22 mm, C = 14 mm, f = 96 Hz. Second line: SW 3 , S = 0.80, D o = 22 mm, C = 14 mm. Left: f = 96 Hz, right: f = 170 Hz. Third line: SW 3 , S = 0.80, D o = 20 mm, C = 10 mm. Left: f = 110 Hz, right: f = 190 Hz. Fourth line: SW 3 , S = 0.60, D o = 18 mm, C = 12 mm. Left: f = 110 Hz, right: f = 180 Hz. The phase is referred to the hot wire signal. Two cycles are represented for better readability.

  f = 110 Hz corresponds to a FTF gain minimum and f = 180 Hz to a FTF gain maximum in Fig.7.1.

Figure 7 . 6 :

 76 Figure 7.6: Q criterion contours deduced from PIV data in cold ow conditions for a forcing level u /ū = 0.30 RMS. The same colorbar, set from 0 to 3. 10 6 s -2 , is applied to all gures. Only positive values of Q are retained. The indicated phase angle is referred to the signal measured by the hot wire.

Figure 7 . 7 :

 77 Figure 7.7: Q criterion contours deduced from PIV data in cold ow conditions for a forcing level u /ū = 0.30 RMS. The same colorbar, set from 0 to 3. 10 6 s -2 , is applied to all gures. Only positive values of Q are retained. The indicated phase angle is referred to the signal measured by the hot wire HW.

Figure 7 . 8 :Figure 7 . 9 :

 7879 Figure 7.8: Details of the injector studied with LES with indication of some sections of interest.

Figure 7 .

 7 Figure 7.10: Examples of two axial cuts used for the determination of velocity proles and Q criterion contours.

Figure 7 .

 7 Figure 7.11: Vorticity (left of symmetry axis) and Q criterion contours (right of symmetry axis) of the cold ow inside the combustion chamber deduced from LES at frequencies corresponding to a minimum FTF gain (f = 120 Hz, top row) and to a maximum FTF gain (f = 190 Hz, bottom row).

Figure 7 .

 7 Figure 7.12: Temporal evolution of axial velocity (red), azimuthal velocity (blue) and pressure (black) signals in sections HW , AS 08mm and AS 56mm . Left: f = 120 Hz. Right: f = 190 Hz.

Figure 7 .

 7 Figure 7.13: Left: evolution of the phase-lags ∆ϕ u θ -u z (full symbols) and ∆ϕ u z -u z (empty symbols) with respect to the distance AS to the swirler backplane. Right: evolution of the convective velocity u c of azimuthal perturbations normalized by the bulk ow velocity u (full symbols) and the maximum velocity u m (empty symbols) reached by the ow in each section as a function of the distance to the swirler backplane. Blue: f = 120 Hz. Red: f = 190 Hz.

Figure 7 .

 7 Figure 7.14: Temporal evolution of axial velocity (red), azimuthal velocity (blue) and swirl number (green) in sections AS 08mm and AS 56mm . Superposed in gray the swirl number deduced from Eq. (5.1). Left: f = 120 Hz. Right: f = 190 Hz.

Figure 7 .

 7 Figure 7.15: Vorticity (left of symmetry axis) and Q criterion contours (right of symmetry axis) from LES at frequencies corresponding to a minimum (f = 120 Hz, top row) and to a maximum FTF gain value (f = 190 Hz, bottom row) in the section S-E upstream the swirler.

Figure 7 .

 7 Figure 7.16: Top: axial velocity proles in section S-E (D o = 22 mm) at frequencies corresponding to a minimum (f = 120 Hz, left) and to a maximum FTF gain value (f = 190 Hz, right). The proles at all the 24 phases are shown for negatives r/R o values. Four characteristic phases are shown for positives r/R o values. The average prole is also superposed as a dashed-dotted line. Bottom: radial gradient of the axial velocity at the channel wall (left) and circulation (right) along the closed contour shown in Fig. 7.15.

Figure 7 .

 7 Figure 7.17: Top: axial velocity proles at the burner outlet section AS 56mm (D o = 20 mm) at frequencies corresponding to a minimum (f = 120 Hz, left) and to a maximum FTF gain value (f = 190 Hz, right). The proles at all the 24 phases are shown for negatives r/R o , while 4 characteristic phases are shown for positives r/R o . The average prole is also superposed as a dash-dotted line. Bottom: radial gradient of the axial velocity at the injector rim (left) and circulation (right) along the closed contour shown in Fig. 7.11.

Figure 7 .

 7 Figure 7.18: Azimuthal (top) and radial (bottom) velocity proles at the burner outlet section AS 56mm (D o = 20 mm) at frequencies corresponding to a minimum (f = 120 Hz, left) and to a maximum FTF gain value (f = 190 Hz, right). The proles at 24 phases in the cycle are shown for negatives r/R o values. Four characteristic phases are shown for positives r/R o values. The average prole is also superposed as a dashed-dotted line.

  An attempt is made to determine dimensionless quantities allowing a scaling of FTF results obtained for dierent injector designs. Following previous studies, a Strouhal number based on the forcing frequency and the ratio between a ame length scale and the convective velocity of vortical perturbations shed in the external shear layer of the swirled jet is considered rst. It is shown that this Strouhal number allows to scale many of the FTF measurements reported in chapter 6. This scaling law is however also shown to fail when the distance δ between the swirler exit and the combustion chamber backplane is modied. A new denition of the Strouhal number is introduced that takes this distance into account which yields a better match to rescale the measured FTFs.

Chapter 8 -

 8 Flame Transfer Function scaling mean ow velocity S L /U b .

Figure 8 . 1 :

 81 Figure 8.1: Sketch of the key mechanisms controlling the FTF.

Figure 8 . 2 :

 82 Figure 8.2: OH* ame light distribution for the dierent congurations tested. First line: eect of nozzle diameter D o . Second line: eect of the injection velocity U b . Third line: eect of the distance δ between swirler exit and combustor backplane.

. 5 )

 5 Figure 8.3: Vortex convection velocity u vc determined from the Q criterion contours shown in Figs. 7.6 and 7.11. Blue: SW 3 , D o = 18 mm, C = 12 mm, f = 180 Hz (PIV). Orange: SW 3 , D o = 20 mm, C = 10 mm, f = 190 Hz (LES). Red: SW 3 , D o = 20 mm, C = 10 mm, f = 190 Hz (PIV). Green: SW 3 , D o = 22 mm, C = 14 mm, f = 170 Hz (PIV).

  1.3-a). Scaled FTF measurements with St b are reported in Fig. 8.4-right. One can see that the inclusion of δ in the denition of the Strouhal number leads to a good superposition of the FTFs obtained for dierent values of the injection diameter D o , bulk ow velocity U b and distance δ between the swirler outlet and the injector outlet section. The minimum gain value appears for all the investigated cases at a Strouhal number St b 1.3-1.4. These results highlight the fact that the timescales of both the mechanisms shown in Fig. 8.1 should be considered for the proper scaling of FTF results of premixed swirling ames.

Figure 8 . 4 :

 84 Figure 8.4: FTF generalization based on Strouhal number. First line: eect of the injector diameter D o . Second line: eect of injection velocity U b . Third line: eect of distance δ between swirler exit and combustor backplane. Left: St a . Right: St b .

  This chapter has been published in the Proceedings of the Combustion Institute with the title Impact of swirl and blu-body on the transfer function of premixed ames. It is reproduced here as is. The objective is to analyze the role of the central blu body on the FTF of swirled ames. To do so, the frequency response of three lean methane/air ames submitted to owrate perturbations is analyzed for ames featuring the same equivalence ratio and thermal power, but a dierent stabilization mechanism. The rst ame is stabilized by a central blu body without swirl, the second one by the same blu body with the addition of swirl and the last one only by swirl, without central insert.In the two last cases, the swirl level is roughly the same. These three ames feature dierent shapes and heat release distributions, but their Flame Transfer Function (FTF) feature about the same phase lag at low frequencies. The gain of the FTF also shows the same behavior for the ame stabilized by the central insert without swirl and the one fully aerodynamically stabilized by swirl. Shedding of vortical structures from the injector nozzle, that grow and rollup the ame tip, controls the FTF of these ames. The ame stabilized by the swirler-plus-blu-body system features a peculiar response, with a large drop of the FTF gain around a frequency at which large swirl number oscillations are observed. Velocity measurements in cold ow conditions reveal a strong reduction of the size of the vortical structures shed from the injector lip at this frequency. The ame stabilized aerodynamically only by swirl and the one stabilized by the blu body without swirl don't exhibit any FTF gain drop at low frequencies. In the former case, large swirl number oscillations are still identied, but large vortical structures shed from the nozzle also persist at the same forcing frequency in the cold ow response. These

Figure 9 . 1 :

 91 Figure 9.1: Experimental setup. S 0 -bb : Non-swirling ame anchored by a blu-body, D 0 = 22 mm, C = 14 mm. S 3 -bb : Flame stabilized by swirl S = 0.8 and the blubody, D 0 = 22 mm, C = 14 mm. S 3 -as : Swirling ame S = 0.75 stabilized aerodynamically, D 0 = 12 mm, β = 15 • . The main dimensions are indicated in millimeters.

Figure 9 . 2 :

 92 Figure 9.2: Gain G and phase lag ϕ of the FTF for the three congurations investigated at the forcing level u /u = 0.30 RMS.

Figure 9 . 4 :

 94 Figure 9.4: Evolutions of the swirl number S and axial velocity u z (r = 0) at the burner outlet with respect to the phase of the hot-wire signal. The relative ame height H /H ref and ame base angle α b /α b oscillations are also plotted. Two oscillation cycles are represented for better readability.

  boundaries i.e. the length of the vertical white segment shown in Fig.9.3 at the phase 195 o . This process proves to be ecient and robust even when the ame is strongly modulated by vortex interaction. The ame angle α b with respect to the vertical direction is also determined at the ame leading edge position as shown in the second sequence in Fig.9.3 at 195 o . Evolutions of H and α b are plotted in Fig. 9.4 over two periods for the three ames S 0 -bb, S 3 -bb and S 3 -as excited at f = 96 Hz. The same reference height H ref = 25 mm is used

Figure 9 . 5 :

 95 Figure 9.5: Q criterion contour obtained from PIV measurements in cold ow conditions, for a forcing level u /u = 0.30 RMS. Isolines of axial velocity are superimposed. Black contour: u z = 0 m/s. Gray contour: u z = -2.5 m/s. The position where the axial velocity u z for the analysis of Fig. 9.4 is measured, is shown as a green cross at the phase 15 o .

Figure A. 1 :

 1 Figure A.1: Setup used for the imaging of naturally unstable (top) or externally excited (bottom) ames.

Figure B. 1 :

 1 Figure B.1: Phase averaged, cold ow, axial PIV scan for two congurations of interest. Top: SW 0 , D o = 22 mm, C = 14 mm, f = 96 Hz. Bottom: SW 3 , D o = 22 mm, C = 14 mm, f = 96 Hz.

Table 1 .

 1 1: Congurations explored. SW: Swirler design: D o : injection tube diameter [mm], C: conical end piece diameter [mm], δ : distance between the swirler exit and the injection plane outlet [mm], U b : velocity at the hot-wire location [m/s]. S * : Estimated swirl number (Eq. (1.5)). S : Swirl number deduced from LDV or PIV measurements.

	SW D o C δ	U b	S
	0	22 14 50 5.44 0.20
	1	18 12 50 5.44 0.54*
	1	18 14 50 5.44 0.48*
	1	20 12 50 5.44 0.63
	1	22 10 50 5.44 0.75*
	1	22 12 50 5.44 0.71
	1	22 14 50 5.44 0.65*
	2	18 12 50 5.44 0.59
	2	18 14 50 5.44 0.52*
	2	20 10 50 5.44 0.73
	2	20 12 50 5.44 0.69
	2	20 14 50 5.44 0.63
	2	22 10 50 5.44 0.84*
	2	22 12 50 5.44 0.79
	3	18 12 50 5.44 0.60
	3	18 14 50 5.44 0.55*
	3	20 10 35 5.44 0.80*
	3	20 10 40 5.44 0.80*
	3	20 10 45 5.44 0.80*
	3	20 10 50 5.44 0.80
	3	20 12 50 2.72 0.73*
	3	20 12 50 4.08 0.73*
	3	20 12 50 5.44 0.73
	3	20 14 50 5.44 0.64*
	3	22 10 50 5.44 0.86*
	3	22 12 50 5.44 0.81
	3	22 14 50 5.44 0.80
	4	22 14 50 5.44 0.95*

Table 3 .

 3 1: NSCBC conditions applied at the boundaries

	Patch	Boundary condition	Relaxation coecient Physical parameters
	Inlet	Top-hat velocity prole	200s -1

Table 4 .

 4 1: Congurations explored. SW: Swirler design: D o : injection tube diameter [mm], C: conical end piece diameter [mm], δ : distance between the swirler exit and the injection plane outlet [mm], U b : bulk ow velocity at the convergent outlet [m/s], A 1 /A 2 : normalized injector cross section area based on hydraulic diameters. S * : Estimated swirl number. S : Swirl number deduced from LDV or PIV measurements.

	SW D o C δ	U b	A 1 /A 2 S
	0	22 14 50 5.44 4.84	0.20
	1	18 12 50 5.44 8.60	0.54*
	1	18 14 50 5.44 13.44	0.48*
	1	20 12 50 5.44 5.36	0.63
	1	22 10 50 5.44 2.86	0.75*
	1	22 12 50 5.44 3.66	0.71
	1	22 14 50 5.44 4.84	0.65*
	2	18 12 50 5.44 8.60	0.59
	2	18 14 50 5.44 13.44	0.52*
	2	20 10 50 5.44 4.00	0.73
	2	20 12 50 5.44 5.36	0.69
	2	20 14 50 5.44 7.56	0.63
	2	22 10 50 5.44 2.86	0.84*
	2	22 12 50 5.44 3.66	0.79
	3	18 12 50 5.44 8.60	0.60
	3	18 14 50 5.44 13.44	0.55*
	3	20 10 35 5.44 4.00	0.80*
	3	20 10 40 5.44 4.00	0.80*
	3	20 10 45 5.44 4.00	0.80*
	3	20 10 50 5.44 4.00	0.80
	3	20 12 50 2.72 5.36	0.73*
	3	20 12 50 4.08 5.36	0.73*
	3	20 12 50 5.44 5.36	0.73
	3	20 14 50 5.44 7.56	0.64*
	3	22 10 50 5.44 2.86	0.86*
	3	22 12 50 5.44 3.66	0.81
	3	22 14 50 5.44 4.84	0.80
	4	22 14 50 5.44 4.84	0.95*
	and that changes in this quantity modify the ame mean and dynamic prop-
	erties. To obtain the swirl number, LDV or PIV measurements of the axial u z
	and azimuthal u

θ velocity components are carried out for the set of geometrical congurations indicated in Tab. 4.1 without star ( * ). The data issued from LDV and PIV measurements are used to determine the swirl number at the CONTENTS 71 injector outlet as:

Table 6 . 1 :

 61 Evolution of the minimum gain level for the dierent congurations explored. SW: Swirler design: D o : injection tube diameter [mm], C: conical end piece diameter [mm], δ : distance between the swirler exit and the injection plane outlet [mm], U b : bulk ow velocity at the convergent outlet [m/s], A 1 /A 2 : normalized injector cross section area based on hydraulic diameters. S * : Estimated swirl number. S : Swirl number deduced from LDV or PIV measurements. G min : minimum gain level. Fig.: corresponding gure reporting the full FTF measurements.

	SW D o	C	δ	U b	A 1 /A 2	S	G min	Fig.
	1	18		50 5.44 8.60	0.54* 1.32	6.8-right
	2	18		50 5.44 8.60	0.59	1.10	6.8-left
	1	20		50 5.44 5.36	0.63	0.93	6.8-right
	2	20		50 5.44 5.36	0.69	0.68	6.8-left, 6.10
	1	22		50 5.44 3.66	0.71	0.55	6.8-right
	2	22		50 5.44 3.66	0.79	0.16	6.8-left
	2	20		50 5.44 7.56	0.63	0.98	6.10
	2	20		50 5.44 4.00	0.73	0.38	6.10
	1	22		50 5.44 4.84	0.65* 0.86	6.12
	3	22		50 5.44 4.84	0.80	0.59	6.12
	4	22		50 5.44 4.84	0.95* 0.38	6.12
	3	20		35 5.44 4.00	0.80* 0.39	6.6-left
	3	20		40 5.44 4.00	0.80* 0.40	6.6-left
	3	20		45 5.44 4.00	0.80* 0.40	6.6-left
	3	20		50 5.44 4.00	0.80	0.37	6.6-left
	3	20		50 2.72 5.36	0.73* 0.80	6.6-right
	3	20		50 4.08 5.36	0.73* 0.76	6.6-right
	3	20		50 5.44 5.36	0.73	0.75	6.6-right

CONTENTS 111

  7.16 are again very close for both excitation frequencies. One begins with the analysis of results obtained for f = 190 Hz. At phase 0 o with respect to the acoustic velocity, the circulation Γ is the lowest. It then linearly grows in the forcing cycle up to phase 75 o . The rate of increase of the circulation Γ then slows down up to a phase lag 180 o . After this stage it remains almost constant up to the end of the forcing cycle. At phase 360 o , one follows the circulation of a new vortical structure.

Table 8 .

 8 1: Parameters used to determine the Strohual numbers St a and St b . δ 1 [mm]: distance between the swirler exit and the central insert of diameter D o , u c,1 [m/s]: convective velocity of u θ along the distance δ 1 , δ 2 [mm]: length of the central insert of diameter D o , u c,2 [m/s]: convective velocity of u θ along the distance δ 2 , L CG [mm]: distance between the injector rim and the ame center of gravity, u vc [m/s]: convective velocity of vortical structures.δ 1 u c,1 δ 2 u c,2 L CG u vc

	D o	18		5.88		9.14 31.0 4.8
	[mm]	20	16	5.88	34	7.23 24.0 4.2
		22		5.88		5.88 21.5 3.8
	U b	2.72		2.94		3.61 22.0 2.10
	[m/s]	4.08	16	4.41	34	5.42 24.5 3.15
		5.44		5.88		7.23 26.0 4.2
		35	1
	δ [mm]	40 45	6 11	5.88 34 7.23 24.0 4.0
		50	16

.6) CONTENTS 153

Chapter 5 -Characterization of thermoacoustic instabilities

Chapter 7 -Flow and flame dynamics

Chapter 9 -Impact of central bluff body

Remerciements

This thesis is part of the CLEAN-Gas European Joint Doctorate and all the work done was possible thanks to the funding received from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 643134. I want to rst thank all the people involved in the organization of the yearly meetings we had over the three years of thesis and particularly Isabella Branca for coordinating the project. and Flame, 142(1):153159. (p. 6) Nicoud, F., Toda, H. B., Cabrit, O., Bose, S., and Lee, J. (2011). Using singular values to build a subgrid-scale model for large eddy simulations. Physics of Fluids, 23 (8). (p. 64) Noiray, N., Durox, D., Schuller, T., and Candel, S. (2008). A unied framework for nonlinear combustion instability analysis based on the ame describing function. J. Fluid. Mech., 615:139167. (p. 7, 8, 96) Oberleithner, K., Schimek, S., and Paschereit, C. O. (2015). Shear ow instabilities in swirl-stabilized combustors and their impact on the amplitude dependent ame response: A linear stability analysis. Combustion and

List of publications Journal articles

Appendix A

Comparison of naturally unstable and externally excited ame motion

Flame transfer function are used in low-order models, in conjunction with descriptions of the burner acoustic eld, to predict stability maps of combustion systems. However, they do not give access to qualitative informations about the ame motion, in response to the acoustic excitation. To analyze the mechanisms behind the ame response, phase-locked images of the ame motion were exploited in this thesis. The experimental setups, used to characterize combustion instabilities and to mimic them by external excitation, are sketched in Fig. A.1. In the rst case, the bottom of the burner was closed by a plate and the exhaust length could be modied by adding tubes of length T = 220 mm. Up to three tubes, 3T = 660 mm, were mounted during this study. Doing so, the thermoacoustic state of the burner was modied and instabilities could be triggered. The uctuating velocity was measured with an hot wire probe and pictures of the ame motion were taken with an ICCD camera, synchronized by the HW signal. In the second case, the exhaust tubes were removed and a loudspeaker was mounted at the bottom of the burner, to introduce the same acoustic excitation measured during self-triggered instability, controlled by the HW signal. Pictures of the ame motion were then taken with the ICCD camera, synchronized by the HW signal.

Comparison of phase-locked ow elds for dierent injector geometries

Steady and dynamic properties of ames stabilized with dierent injector geometries, were analyzed in this work. The mechanisms behind the response to acoustic perturbations of these ames were studied with dierent diagnostics, including PIV measurements. The main objective was to investigate the conditions leading to an high or low ame response at dierent frequencies. The dierent strength of vortical structures, generated at the burner rim and then convected in the combustion chamber, was individuated as one of the peculiar mechanisms. This strength was estimated by analyzing the Q criterion contour plots, issued from PIV data as:

(B.1)

The PIV scans, used to derive the Q criterion plots, are shown in Abstract : Lean premixed (LPM) combustion systems achieve low pollutant emission levels, with compact flames and high power densities, but are highly sensitive to dynamic phenomena, e.g, flashback, blowout and thermoacoustic instabilities, that hinder their practical application. Most LPM gas turbine combustors use swirling flows to stabilize compact flames for efficient and clean combustion. A better knowledge of the mechanisms of steady and unsteady combustion of lean premixed swirled mixtures is then of practical, as well as fundamental interest. This thesis is a contribute towards the achievement of this goal. A burner, made of several components with variable geometry, was specifically designed for this scope. An experimental analysis was conducted to investigate the main parameters leading to a reduction of the sensitivity of LPM systems to dynamic phenomena. The diagnostics applied include flame transfer function (FTF) measurements, laser diagnostics (LDV and PIV) and flame imaging. Large eddy simulations were also exploited to elucidate the mechanisms behind the experimental observations.

Universit é Paris-Saclay

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