Dissertations / Theses on the topic 'Dynamic Mode Decomposition (DMD)'

Combustion instability due to feedback coupling between unsteady heat release and natural acoustic modes can cause catastrophic failure in liquid rocket engines and to predict and prevent these instabilities the mechanisms that drive them must be further elucidated. With this goal in mind, the objective of this thesis was to develop techniques that improve the understanding of the specific underlying physical processes involved in these driving mechanisms. In particular, this work sought to develop a small-scale, optically accessible liquid rocket engine simulator and to apply modern, high-speed diagnostic techniques to characterize the reacting flow and acoustic field within the simulator. Specifically, high-speed (10 kHz), simultaneous data were acquired while the simulator was experiencing a 170 Hz combustion instability using particle image velocimetry, OH planar laser induced fluorescence, CH* chemiluminescence, and dynamic pressure measurements. In addition, this work sought to develop approaches to reduce the large quantities of data acquired, extracting key physical phenomena involved in the driving mechanisms. The initial data reduction approach was chosen based on the fact that the combustion instability problem is often simplified to the point that it can be characterized by an approximately linear constant coefficient system of equations. Consistent with this simplification, the experimental data were analyzed by the dynamic mode decomposition method. The developed approach to apply the dynamic mode decomposition to simultaneously acquired data located a coupled hydrodynamic/combustion/acoustic mode at 1017 Hz. On the other hand, the dynamic mode decomposition's assumed constant operator approach failed to locate any modes of interest near 170 Hz. This led to the development of two new data analysis techniques based on the dynamic mode decomposition and Floquet theory that assume that the experiment is governed by a linear, periodic system of equations. The new periodic-operator data analysis techniques, the Floquet decomposition and the ensemble Floquet decomposition, approximate, from experimental data, the largest moduli Floquet multipliers, which determine the stability of the periodic solution trajectory of the system. The unstable experiment dataset was analyzed with these techniques and the ensemble Floquet decomposition analysis found a large modulus Floquet multiplier and associated mode with a frequency of 169.6 Hz. Furthermore, the approximate Rayleigh criterion indicated that this mode was unstable with respect to combustion instability. Overall, based on the positive finding that the ensemble Floquet decomposition was able to locate an unstable combustion mode at 170 Hz when the operator's time period was set to 1 ms, suggests that the dynamic mode decomposition based 1017 Hz mode parametrically forces the 170 Hz mode, resulting in what could be characterized as a parametric combustion instability.

La combustion prévaporisée prémélangée pauvre est une piste de choix pour réduire les émissions polluantes des moteurs d'avions mais peut conduire à l'apparition d'instabilités thermo-acoustiques. Afin d'améliorer la stabilité de telles flammes, l'étagement du combustible consiste à contrôler la distribution spatiale du carburant. Une telle procédure s'accompagne cependant d'une complexité accrue du système pouvant déboucher sur des phénomènes inattendus.Un brûleur à l'échelle de laboratoire alimenté par du dodécane liquide est utilisé dans cette thèse. Le combustible est injecté dans deux étages séparés, permettant ainsi de contrôler sa distribution. Cette particularité permet l'observation de différentes formes de flammes et notamment de points bistables pour lesquels deux flammes différentes peuvent exister malgré des conditions opératoires identiques.L'utilisation de diagnostics optiques à haute cadence (diffusion de Mie des gouttes de combustible et émission spontanée de la flamme) est couplée à des méthodes de post-traitement avancées comme la Décomposition en Modes Dynamiques. Ainsi, des mécanismes pilotant la stabilisation des flammes ainsi que leurs changements de forme sont proposés. Ils mettent notamment en lumière les interactions entre l'écoulement gazeux, les gouttes de combustible et la flamme
A promising way to reduce jet engines pollutant emissions is the use of lean premixed prevaporized combustion but it tends to trigger thermo-acoustic instabilities. To improve the stability of these flames, a procedure called staging consists in splitting the fuel injection to control its spatial distribution. This however leads to an increased complexity and unexpected phenomena can occur.In the present work, a model gas turbine combustor fed with liquid dodecane is used. It is equipped with two fuel injection stages to control the fuel distribution in the burner. Different flame stabilizations can be observed and a bistable case where two flame shapes can exist for the same operating conditions is highlighted.High-speed optical diagnostics (fuel droplets Mie scatering and chemiluminescence measurements) are coupled with advanced post-processing methods like Dynamic Mode Decomposition. The results enable to propose mechanisms leading to flame stabilization and flame shape transitions. They show a strong interplay between the gaseous flow, the fuel droplets and the flame itself

Depuis l'identification par Brown \& Roshko, en 1974, de structures jouant un rôle majeur dans le mélange d'un écoulement turbulent, la recherche de structures cohérentes a été un des principaux axes d'étude en mécanique des fluides.Les travaux présentés dans ce manuscrit s'inscrivent dans cette voie.La première partie du manuscrit traite ainsi de l'identification de structures cohérentes. Elle se compose de trois chapitres abordant deux techniques d'identification. La Décomposition en mode dynamique (DMD), ainsi que des variantes généralisant son champ applicatif est présenté dans le premier chapitre. Cette méthode propose une représentation par modes spatiaux et temporels d'un ensemble de données. Une méthode pour la sélection de composantes particulièrement représentatives de la dynamique, i.e. présentant de bonnes qualités d'observabilité, se basant sur cette décomposition est également décrite dans ce chapitre.Le deuxième chapitre traite de la détection de structures cohérentes lagrangienne, par suivi de particules. Ces structures permettent d'identifier les frontières matérielles et apportent des éclaircissements sur les mécanismes du mélange au sein de l'écoulement considéré.Ces méthodes sont appliquées, dans le chapitre trois, au cas d'un écoulement incompressible affleurant une cavité ouverte.La seconde partie du manuscrit traite des questions de représentation et discrimination de données scientifiques.Une réponse à la question de la représentation de structures cohérentes a été la mise en place d'outils permettant la visualisation interactive de jeux de données scientifiques, qui dont la présentation fait l'objet du chapitre quatre. En particulier, l'utilisation d'objets tangibles, représentant les données dans le monde réel, permet une exploration plus efficace des ensembles de volumétriques de données scientifiques. La question d'une perception et discrimination efficace de données représentées, e.g. la différentiation entre deux valeurs proches, est abordée dans le cinquième chapitre.

Lors d’un vol hypersonique (Mach 6, 20 km d’altitude) la couche limite se développant sur l’avant-corps d’un véhicule hypersonique est laminaire. Cet état cause un désamorçage du moteur (statoréacteur) assurant la propulsion du véhicule. Pour pallier ce problème, il faut forcer la transition de la couche limite á l’aide d’un dispositif de contrôle dont l’effet est permanent (passif) ou modulable (actif) pendant le vol. Dans ce travail, nous analysons l’efficacité d’un dispositif actif d’injection d’air á la paroi pour forcer la transition de la couche limite sur un avant-corps générique. L’interaction jet d’air/couche limite est simulée numériquement avec une approche aux grandes échelles (LES). Une étude paramétrique sur la pression d’injection permet de quantifier l’efficacité du jet á déstabiliser la couche limite. L’influence des conditions de vol (altitude, Mach) sur la transition est également étudiée. Une analyse des résultats de simulation par Décomposition en Modes Dynamiques (DMD) est menée pour comprendre quels sont les modes dynamiques responsables de la transition et les mécanismes sous-jacents. Des essais dans la soufflerie silencieuse de l’université de Purdue (BAM6QT) ont été effectués pour tester expérimentalement l’efficacité des dispositifs passifs (rugosité isolée en forme de losange) et actifs (mono-injection d’air) pour faire transitionner la couche limite. Une peinture thermo-sensible et des capteurs de pression (PCB, Kulite) ont été utilisés pour déterminer la nature de la couche limite. Les résultats de ce travail montrent qu’une injection sonique suffit pour forcer la couche limite. On observe des essais, que pour une même hauteur de pénétration, les rugosités isolées sont moins efficaces que les jets (mono injection) pour déstabiliser la couche limite
During a hypersonic flight (Mach 6, 20 km altitude), the boundary layer developing on the forebody of a vehicle is laminar. This state may destabilize the scramjet engine propelling the vehicle. To overcome this problem during the flight, the boundary layer transition has to be forced using a control device whose effect is fixed (passive) or adjustable (active). In this work, we analyze the efficiency of a jet in crossflow in forcing the boundary layer transition on a generic forebody. The flow is computed with a Large Eddy Simulations (LES) approach. A parametric study of the injection pressure allows the efficiency of the jet in tripping the boundary layer to be quantified. The influence of flight conditions (Mach, altitude) on the transition is also studied. Dynamic Mode Decomposition (DMD) is applied to the simulation results to determine the transition leading to dynamic modes and to understand underlying transition mechanisms. Experiments in the Purdue University quiet wind tunnel (BAM6QT) were performed to quantify the efficiency of a passive transition device (diamond roughnesses) and an active transition device (single air jet) in tripping the boundary layer. A thermo-sensitive paint and pressure transducers (Kulite, PCB) were used to determine the state of the boundary layer on the generic forebody. Experimental and numerical results show a sonic injection is sufficient to induce transition. We observe from the experiments that for the same penetration height, a single roughness is less efficient than a single air jet in destabilizing the boundary layer

The method of Dynamic Mode Decomposition (DMD) was introduced originally in the area of Computatational Fluid Dynamics (CFD) for extracting coherent structures from spatio-temporal complex fluid flow data. DMD takes in time series data and computes a set of modes, each of which is associated with a complex eigenvalue. DMD analysis is closely associated with spectral analysis of the Koopman operator, which provides linear but infinite-dimensional representation of nonlinear dynamical systems. Therefore, by using DMD a nonlinear system could be described by a superposition of modes whose dynamics are governed by the eigenvalues. The key advantage of DMD is its data-driven nature which does not rely on any prior assumptions except the inherent dynamics which are observed over time. Its capability for extracting relevant modes from complex fluid flows has seen significant application across multiple fields, including computer vision, robotics and neuroscience. This thesis, in order to expand DMD to other applications, advances the original formulation so that it can be used to solve novel problems in the fields of signal processing and computer vision. In signal processing this thesis introduces the method of using DMD for decomposing a univariate time series into a number of interpretable elements with different subspaces, such as noise, trends and harmonics. In addition, univariate time series forecasting is shown using DMD. The computer vision part of this thesis focuses on innovative applications pertaining to the areas of medical imaging, biometrics and background modelling. In the area of medical imaging a novel DMD framework is proposed that introduces windowed and reconstruction variants of DMD for quantifying kidney function in Dynamic Contrast Enhanced Magnetic Resonance imaging (DCE-MRI) sequences, through movement correction and functional segmentation of the kidneys. The biometrics portion of this thesis introduces a DMD based classification pipeline for counter spoofing 2D facial videos and static finger vein images. The finger vein counter spoofing makes use of a novel atemporal variant of DMD that captures micro-level artefacts that can differentiate the quality and light reflection properties between a live and a spoofed finger vein image, while the DMD on 2D facial image sequences distinguishes attack specific cues from a live face by capturing complex dynamics of head movements, eye-blinking and lip-movements in a data driven manner. Finally, this thesis proposes a new technique using DMD to obtain a background model of a visual scene in the colour domain. These aspects form the major contributions of this thesis. The results from this thesis present DMD as a promising approach for applications requiring feature extraction including: (i) trends and noise from signals, (ii) micro-level texture descriptor from images, and (iii) coherent structures from image sequences/videos, as well as applications that require suppression of movements from dynamical spatio-temporal image sequences.

The human brain is the most important organ of the human body. It controls our thoughts, movements and emotions. For that reason, protecting the brain from harm is of the utmost importance but to protect the brain, one must first understand brain injuries. Currently, observations and criteria involving brain injury are focused around acceleration and forces. However, the brain is poorly understood in the frequency domain. This study uses finite element analysis to simulate impact for 5 different impact angles. Then a numerical technique called dynamic mode decomposition is used to extract modal properties for brain tissue in regions near the corpus callosum and brain stem. Three modal frequencies were identified with frequency ranges of (44-68) Hz, (68-155) Hz, and (114-299) Hz. Additionally, it was observed that impact angle, displacement direction, and region of the brain have a significant impact on the modal response of brain tissue during impact.

Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.
Master of Science
The Navier-Stokes (NS) equations are the primary mathematical model for understanding the behavior of fluids. The existence and smoothness of the NS equations is considered to be one of the most important open problems in mathematics, and challenges in their numerical simulation is a barrier to understanding the physical phenomenon of turbulence. Due to the difficulty of studying this problem directly, simpler problems in the form of nonlinear partial differential equations (PDEs) that exhibit similar properties to the NS equations are studied as preliminary steps towards building a wider understanding of the field. Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations. Naturally, reduced-order modeling techniques come at the price of either computational accuracy or computation time. Optimization techniques are studied to improve either or both of these objectives and decrease the total computational cost of the problem. This thesis focuses on the dynamic mode decomposition (DMD) applied to nonlinear PDEs with periodic boundary conditions. It provides one study of an existing optimization framework for the DMD method known as the Optimized DMD and provides another study of a newly proposed optimization framework for the DMD method called the Split DMD.

On s’intéresse dans cette thèse à l’apprentissage d’un modèle réduit précis et stable, à partir de données correspondant à la solution d’une équation aux dérivées partielles (EDP), et générées par un solveur haute fidélité (HF). Pour ce faire, on utilise la méthode Dynamic Mode Decomposition (DMD) ainsi que la méthode de réduction Proper Orthogonal Decomposition (POD). Le modèle réduit appris est facilement interprétable, et par une analyse spectrale a posteriori de ce modèle on peut détecter les anomalies lors de la phase d’apprentissage. Les extensions au cas de couplage EDP-EDO, ainsi qu’au cas d’EDP d’ordre deux en temps sont présentées. L’apprentissage d’un modèle réduit dans le cas d’un système dynamique contrôlé par commutation, où la règle de contrôle est apprise à l’aide d’un réseau de neurones artificiel (ANN), est également traité. Un inconvénient de la réduction POD, est la difficile interprétation de la représentation basse dimension. On proposera alors l’utilisation de la méthode Empirical Interpolation Method (EIM). La représentation basse dimension est alors intelligible, et consiste en une restriction de la solution en des points sélectionnés. Cette approche sera ensuite étendue au cas d’EDP dépendant d’un paramètre, et où l’algorithme Kernel Ridge Regression (KRR) nous permettra d’apprendre la variété solution. Ainsi, on présentera l’apprentissage d’un modèle réduit paramétré. L’extension au cas de données bruitées ou bien au cas d’EDP d’évolution non linéaire est présentée en ouverture
In this thesis, an algorithm for learning an accurate reduced order model from data generated by a high fidelity solver (HF solver) is proposed. To achieve this goal, we use both Dynamic Mode Decomposition (DMD) and Proper Orthogonal Decomposition (POD). Anomaly detection, during the learning process, can be easily done by performing an a posteriori spectral analysis on the reduced order model learnt. Several extensions are presented to make the method as general as possible. Thus, we handle the case of coupled ODE/PDE systems or the case of second order hyperbolic equations. The method is also extended to the case of switched control systems, where the switching rule is learnt by using an Artificial Neural Network (ANN). The reduced order model learnt allows to predict time evolution of the POD coefficients. However, the POD coefficients have no interpretable meaning. To tackle this issue, we propose an interpretable reduction method using the Empirical Interpolation Method (EIM). This reduction method is then adapted to the case of third-order tensors, and combining with the Kernel Ridge Regression (KRR) we can learn the solution manifold in the case of parametrized PDEs. In this way, we can learn a parametrized reduced order model. The case of non-linear PDEs or disturbed data is finally presented in the opening

Historic rock-mounted lighthouses play a vital role in the safe navigation around perilous reefs. Their longevity is threatened by the battering of waves which may be set to increase with climate change. The protection of this historic heritage needs the identification of both structural dynamic parameters (natural frequencies and shape modes), and of the worst-cases wave load combination, able to affect that natural frequencies. This dissertation was developed during a period of five months at University of Plymouth, along with the researching team of the project STORMLAMP. The project is divided in three parts; the first involving a meteocean analysis, developed by means of peak over threshold technique, aimed to address realistically a test campaign held afterwards. The second, focused on the dynamic analysis of the Dubh Artach lighthouse and was developed by means of a Matlab toolbox provided to the group by Prof. Brownjohn from Exeter University, partner of the project as well. It is aimed on one hand to detect the dynamic properties of the structure and, on the other hand, to recognize eventual directionality in the structural response. The third phase was held at Plymouth University laboratory “COAST”. During this phase, a laboratory campaign, involving more than 100 tests, allowed to perform a parametric analysis aimed to identify the parameters, of an extreme wave, that influence more the impact force and that the wave exerts on the structure. To extract impact time history, force signals were decomposed by means of Empirical mode decomposition and Duhamel integral algorithms. Image analysis, moreover, allowed to locate run-up caused by those waves upon a steel cylinder and to integrate a study of the run-up as well. The analysis led to several considerations useful on one hand for the prediction of the worst-case loading of the Dubh Artach lighthouse and, on the other hand, for the introduction of the NewWave theory for the design of coastal structures.

The study concerns the analysis of vocal cycle length perturbations in normophonic and dysphonic speakers.A method for tracking cycle lengths in voiced speech is proposed. The speech cycles are detected via the saliences of the speech signal samples, defined as the length of the temporal interval over which a sample is a maximum. The tracking of the cycle lengths is based on a dynamic programming algorithm that does not request that the signal is locally periodic and the average period length known a priori.The method is validated on a corpus of synthetic stimuli. The results show a good agreement between the extracted and the synthetic reference length time series. The method is able to track accurately low-frequency modulations and ast cycle-to-cycle perturbations of up to 10% and 4% respectively over the whole range of vocal frequencies. Robustness with regard to the background noise has lso been tested. The results indicate that the tracking is reliable for signal-to-noise ratios higher than 15dB.A method for analyzing the size of the cycle length perturbations as well as their frequency is proposed. The cycle length time series is decomposed into a sum of oscillating components by empirical mode decomposition the instantaneous envelopes and frequencies of which are obtained via AM-FM decomposition. Based on their average instantaneous frequencies, the empirical modes are then assigned to four categories (declination, physiological tremor, neurological tremor as well as cycle length jitter) and added within each. The within-category size of the cycle length perturbations is estimated via the standard deviation of the empirical mode sum divided by the average cycle length. The neurological tremor modulation frequency and bandwidth are obtained via the instantaneous frequencies and amplitudes of empirical modes in the neurological tremor category and summarized via a weighted instantaneous frequency probability density, compensating for the effects of mode mixing.The method is applied to two corpora of vowels comprising 123 and 74 control and 456 and 205 Parkinson speaker recordings respectively. The results indicate that the neurological tremor modulation depth is statistically significantly higher for female Parkinson speakers than for female control speakers. Neurological tremor frequency differs statistically significantly between male and female speakers and increases statistically significantly for the pooled Parkinson speakers compared to the pooled control speakers. Finally, the average vocal frequency increases for male Parkinson speakers and decreases for female Parkinson speakers, compared to the control speakers.
Doctorat en Sciences de l'ingénieur et technologie
info:eu-repo/semantics/nonPublished

La simulation aux grandes échelles est devenue un outil d’analyse incontournable pour l’étude des écoulements turbulents dans des géométries complexes. Cependant, à cause de l’augmentation constante des ressources de calcul, le traitement des grandes quantités de données générées par les simulations hautement résolues est devenu un véritable défi qu’il n’est plus possible de relever avec des outils traditionnels. En mécanique des fluides numérique, cette problématique émergente soulève les mêmes questions que celles communément rencontrées en informatique avec des données massives. A ce sujet, certaines méthodes ont déjà été développées telles que le partitionnement et l’ordonnancement des données ou bien encore le traitement en parallèle mais restent insuffisantes pour les simulations numériques modernes. Ainsi, l’objectif de cette thèse est de proposer de nouveaux formalismes permettant de contourner le problème de volume de données en vue des futurs calculs exaflopiques que l’informatique devrait atteindre en 2020. A cette fin, une méthode massivement parallèle de co-traitement, adaptée au formalisme non-structuré, a été développée afin d’extraire les grandes structures des écoulements turbulents. Son principe consiste à introduire une série de grilles de plus en plus grossières réduisant ainsi la quantité de données à traiter tout en gardant intactes les structures cohérentes d’intérêt. Les données sont transférées d’une grille à une autre grâce à l’utilisation de filtres et de méthodes d’interpolation d’ordre élevé. L’efficacité de cette méthodologie a pu être démontrée en appliquant des techniques de décomposition modale lors de la simulation 3D d’une pale de turbine turbulente sur une grille de plusieurs milliards d’éléments. En outre, cette capacité à pouvoir gérer plusieurs niveaux de grilles au sein d’une simulation a été utilisée par la suite pour la mise en place de calculs basés sur une stratégie multi-niveaux. L’objectif de cette méthode est d’évaluer au cours du calcul les erreurs numériques et celles liées à la modélisation en simulant simultanément la même configuration pour deux résolutions différentes. Cette estimation de l’erreur est précieuse car elle permet de générer des grilles optimisées à travers la construction d’une mesure objective de la qualité des grilles. Ainsi, cette méthodologie de multi-résolution tente de limiter le coût de calcul de la simulation en minimisant les erreurs de modélisation en sous-maille, et a été appliquée avec succès à la simulation d’un écoulement turbulent autour d’un cylindre
Large-Eddy Simulation (LES) has become a major tool for the analysis of highly turbulent flows in complex geometries. However, due to the steadily increase of computational resources, the amount of data generated by well-resolved numerical simulations is such that it has become very challenging to manage them with traditional data processing tools. In Computational Fluid Dynamics (CFD), this emerging problematic leads to the same "Big Data" challenges as in the computer science field. Some techniques have already been developed such as data partitioning and ordering or parallel processing but still remain insufficient for modern numerical simulations. Hence, the objective of this work is to propose new processing formalisms to circumvent the data volume issue for the future 2020 exa-scale computing objectives. To this aim, a massively parallel co-processing method, suited for complex geometries, was developed in order to extract large-scale features in turbulent flows. The principle of the method is to introduce a series of coarser nested grids to reduce the amount of data while keeping the large scales of interest. Data is transferred from one grid level to another using high-order filters and accurate interpolation techniques. This method enabled to apply modal decomposition techniques to a billion-cell LES of a 3D turbulent turbine blade, thus demonstrating its effectiveness. The capability of performing calculations on several embedded grid levels was then used to devise the multi-resolution LES (MR-LES). The aim of the method is to evaluate the modeling and numerical errors during an LES by conducting the same simulation on two different mesh resolutions, simultaneously. This error estimation is highly valuable as it allows to generate optimal grids through the building of an objective grid quality measure. MR-LES intents to limit the computational cost of the simulation while minimizing the sub-grid scale modeling errors. This novel framework was applied successfully to the simulation of a turbulent flow around a 3D cylinder

Le contrôle d'écoulements turbulents est un enjeu majeur en aérodynamique. Cependant, la présence d'un grand nombre de degrés de libertés et d'une dynamique complexe rend délicat la modélisation dynamique de ces écoulements qui est pourtant nécessaire à la conception d'un contrôle efficace. Au cours de cette thèse, différentes directions ont été suivies afin de développer des modèles réduits dans des configurations réalistes d'écoulements et d'utiliser ces modèles pour le contrôle.Premièrement, la décomposition en modes dynamiques (DMD), et certaines de ses variantes, ont été exploitées en tant que base réduite afin d'extraire au mieux le comportement dynamique de l'écoulement. Par la suite, nous nous sommes intéressés à l'assimilation de données 4D-Var qui permet de combiner des informations inhomogènes provenant d'un modèle dynamique, d'observations et de connaissances a priori du système. Nous avons ainsi élaboré des modèles réduits POD et DMD d'un écoulement turbulent autour d'un cylindre à partir de données expérimentales PIV. Finalement, nous avons considéré le contrôle d'écoulement dans un contexte d'interaction fluide/structure. Après avoir montré que les mouvements de solides immergés dans le fluide pouvaient être représentés comme une contrainte supplémentaire dans le modèle réduit, nous avons stabilisé un écoulement de sillage de cylindre par oscillation verticale
Control of turbulent flows is still today a challenge in aerodynamics. Indeed, the presence of a high number of active degrees of freedom and of a complex dynamics leads to the need of strong modelling efforts for an efficient control design. During this PhD, various directions have been followed in order to develop reduced-order models of flows in realistic situations and to use it for control. First, dynamic mode decomposition (DMD), and some of its variants, have been exploited as reduced basis for extracting at best the dynamical behaviour of the flow. Thereafter, we were interested in 4D-variational data assimilation which combines inhomogeneous informations coming from a dynamical model, observations and an a priori knowledge of the system. POD and DMD reduced-order models of a turbulent cylinder wake flow have been successfully derived using data assimilation of PIV measurements. Finally, we considered flow control in a fluid-structure interaction context. After showing that the immersed body motion can be represented as an additional constraint in the reduced-order model, we stabilized a cylinder wake flow by vertical oscillations

La nécessité d’une meilleure compréhension du mécanisme d’entrainement pour l’instabilité à basse fréquence observée dans un écoulement dans une tuyère sur-détendue a été discutée. Le caractère instable de l’onde de choc/couche limite reste un défi pratique important pour les problèmes des écoulements dans les tuyères. De plus, pour une couche limite turbulente incidente donnée, ce type d’écoulement présente généralement des mouvements de choc à basse fréquence plus élevées qui sont moins couplés aux échelles de temps de la turbulence en amont. Cela peut être bon du point de vue d’un expérimentateur, en raison de difficultés à mesurer des fréquences plus élevées, mais c’est plus difficile d’un point de vue calcul numérique en raison de la nécessité d’obtenir des séries temporelles plus longues pour résoudre les mouvements à basse fréquence. En excellent accord avec les résultats expérimentaux, une série de calcul LES de très longue durée a été réalisée, il a été clairement démontré l’existence de mouvements énergétiques à basse fréquence et à large bande près du point de séparation. Des efforts particuliers ont été faits pour éviter tout forçage à basse fréquence en amont, et il a été explicitement démontré que les oscillations de choc à basse fréquence observées n’étaient pas liées à la génération de turbulence d’entrée, excluant la possibilité d’un artefact numérique. Différentes méthodes d’analyse spectrales, et en décomposition en mode dynamique ont été utilisées pour montrer que les échelles de temps impliquées dans un tel mécanisme sont environ deux ordres de grandeur plus grandes que les échelles de temps impliquées dans la turbulence de la couche limite, ce qui est cohérent avec les mouvements de basse fréquence observés. En outre, ces échelles de temps se sont avérées être fortement modulées par la quantité de flux inversé à l’intérieur de la bulle de séparation. Ce scénario peut, en principe, expliquer à la fois l’instabilité des basses fréquences et sa nature à large bande
The need for a better understanding of the driving mechanism for the observed low-frequency unsteadiness in an over-expanded nozzle flows was discussed. The unsteady character of the shock wave/boundary layer remains an important practical challenge for the nozzle flow problems. Additionally, for a given incoming turbulent boundary layer, this kind of flow usually exhibits higher low-frequency shock motions which are less coupled from the timescales of the incoming turbulence. This may be good from an experimenter’s point of view, because of the difficulties in measuring higher frequencies, but it is more challenging from a computational point of view due to the need to obtain long time series to resolve low-frequency movements. In excellent agreement with the experimental findings, a very-long LES simulation run was carried out to demonstrate the existence of energetic broadband low-frequency motions near the separation point. Particular efforts were done in order to avoid any upstream low-frequency forcing, and it was explicitly demonstrated that the observed low-frequency shock oscillations were not connected with the inflow turbulence generation, ruling out the possibility of a numerical artefact. Different methods of spectral analysis and dynamic mode decomposition have been used to show that the timescales involved in such a mechanism are about two orders of magnitude larger than the time scales involved in the turbulence of the boundary layer, which is consistent with the observed low-frequency motions. Furthermore, those timescales were shown to be strongly modulated by the amount of reversed flow inside the separation bubble. This scenario can, in principle, explain both the low-frequency unsteadiness and its broadband nature

La déformation d’une capsule en écoulement dans un canal micro-fluidique est un problème compliqué à simuler numériquement. Nous proposons deux modèles innovants de pilotage de données d’ordre réduit pour simuler le problème spatio-temporel à partir d’une base de données collectée des simulations réalisées avec un modèle d’ordre élevé. L’objectif est de remplacer le modèle numérique haute-fidélité existant par un modèle d’ordre réduit capable de simuler l’évolution de déformation des capsules en écoulement à faible cout en temps et en calcul. Le premier modèle consiste à construire à partir d’un cube de données espace-temps-paramètre un modèle réduit pour simuler la déformation de la microcapsule pour n’importe quelle configuration admissible de paramètres. La prédiction de l’évolution temporelle de la capsule pour une configuration donnée de paramètres et un pas de discrétisation temporelle choisi se fait à l’aide d’un apprentissage sur des variétés du modèle réduit. Le deuxième modèle se base sur l’idée de réécrire le problème sous forme d’un système dynamique d’ordre réduit dans lequel les coefficients spectraux des déplacements et les champs des vitesses sont relies à travers d’un opérateur dynamique à identifier. Pour déterminer ce dernier, nous suggérons l’utilisation d’une approche de décomposition en modes dynamiques. Des validations numériques confirment la fiabilité et stabilité des deux nouveaux modèles par rapport au modèle d’ordre élevé. Une application informatique est également mise au point afin d’explorer l’évolution de déformation des capsules pour toute configuration de paramètres admissibles
The motion of a liquid-filled microcapsule flowing in a microchannel is a complex problem tosimulate. Two innovative reduced-order data-driven models are proposed to replace the Fluid Structure Interaction (FSI) model using a collected database from high-fidelity simulations. The objective is to replace the existing Full Order Model (FOM) with a fast-simulation model that can simulate the capsule deformation in flow at a low cost in terms of time and calculation. The first model consists in building from a space-time-parameter datacube a reduced model to simulate the deformation of the microcapsule for any admissible configuration of parameters. Time evolution of the capsule deformation is treated by identifying the nonlinear low-order manifold of the reduced variables. Then, manifold learning is applied using the Diffuse Approximation (DA) method to predict capsule deformation for a query configuration of parameters and a chosen time discretization. The second model is based on rewriting the FSI model under the form of a reduced-order dynamic system. In this latter, the spectral displacement and velocity coefficients are related through a dynamic operator to be identified. To determine this operator, we suggest the use of a dynamic mode decomposition approach. Numerical validations prove the reliability and stability of the two new models compared to the high order model. A software application has been developed to explore the capsule deformation evolution for any couple of admissible parameters

[ES] El principal desafío en los motores turbina de gas empleados en aviación reside en aumentar la eficiencia del ciclo termodinámico manteniendo las emisiones contaminantes por debajo de las rigurosas restricciones. Ésto ha conllevado la necesidad de diseñar nuevas estrategias de inyección/combustión que operan en puntos de operación peligrosos por su cercanía al límite inferior de apagado de llama. En este contexto, el concepto Lean Direct Injection (LDI) ha emergido como una tecnología prometedora a la hora de reducir los óxidos de nitrógeno (NOx) emitidos por las plantas propulsoras de los aviones de nueva generación. En este contexto, la presente tesis tiene como objetivos contribuir al conocimiento de los mecanismos físicos que rigen el comportamiento de un quemador LDI y proporcionar herramientas de análisis para una profunda caracterización de las complejas estructuras de flujo de turbulento generadas en el interior de la cámara de combustión. Para ello, se ha desarrollado una metodología numérica basada en CFD capaz de modelar el flujo bifásico no reactivo en el interior de un quemador LDI académico mediante enfoques de turbulencia U-RANS y LES en un marco Euleriano-Lagrangiano. La resolución numérica de este problema multi-escala se aborda mediante la descripción completa del flujo a lo largo de todos los elementos que constituyen la maqueta experimental, incluyendo su paso por el swirler y entrada a la cámara de combustión. Ésto se lleva a cabo través de dos códigos CFD que involucran dos estrategias de mallado diferentes: una basada en algoritmos de generación y refinamiento automático de la malla (AMR) a través de CONVERGE y otra técnica de mallado estático más tradicional mediante OpenFOAM. Por un lado, se ha definido una metodología para obtener una estrategia de mallado óptima mediante el uso del AMR y se han explotado sus beneficios frente a los enfoques tradicionales de malla estática. De esta forma, se ha demostrado que la aplicabilidad de las herramientas de control de malla disponibles en CONVERGE como el refinamiento fijo (fixed embedding) y el AMR son una opción muy interesante para afrontar este tipo de problemas multi-escala. Los resultados destacan una optimización del uso de los recursos computacionales y una mayor precisión en las simulaciones realizadas con la metodología presentada. Por otro lado, el uso de herramientas CFD se ha combinado con la aplicación de técnicas de descomposición modal avanzadas (Proper Orthogonal Decomposition and Dynamic Mode Decomposition). La identificación numérica de los principales modos acústicos en la cámara de combustión ha demostrado el potencial de estas herramientas al permitir caracterizar las estructuras de flujo coherentes generadas como consecuencia de la rotura de los vórtices (VBB) y de los chorros fuertemente torbellinados presentes en el quemador LDI. Además, la implementación de estos procedimientos matemáticos ha permitido tanto recuperar información sobre las características de la dinámica de flujo como proporcionar un enfoque sistemático para identificar los principales mecanismos que sustentan las inestabilidades en la cámara de combustión. Finalmente, la metodología validada ha sido explotada a través de un Diseño de Experimentos (DoE) para cuantificar la influencia de los factores críticos de diseño en el flujo no reactivo. De esta manera, se ha evaluado la contribución individual de algunos parámetros funcionales (el número de palas del swirler, el ángulo de dichas palas, el ancho de la cámara de combustión y la posición axial del orificio del inyector) en los patrones del campo fluido, la distribución del tamaño de gotas del combustible líquido y la aparición de inestabilidades en la cámara de combustión a través de una matriz ortogonal L9 de Taguchi. Este estudio estadístico supone un punto de partida para posteriores estudios de inyección, atomización y combus
[CA] El principal desafiament als motors turbina de gas utilitzats a la aviació resideix en augmentar l'eficiència del cicle termodinàmic mantenint les emissions contaminants per davall de les rigoroses restriccions. Aquest fet comporta la necessitat de dissenyar noves estratègies d'injecció/combustió que radiquen en punts d'operació perillosos per la seva aproximació al límit inferior d'apagat de flama. En aquest context, el concepte Lean Direct Injection (LDI) sorgeix com a eina innovadora a l'hora de reduir els òxids de nitrogen (NOx) emesos per les plantes propulsores dels avions de nova generació. Sota aquest context, aquesta tesis té com a objectius contribuir al coneixement dels mecanismes físics que regeixen el comportament d'un cremador LDI i proporcionar ferramentes d'anàlisi per a una profunda caracterització de les complexes estructures de flux turbulent generades a l'interior de la càmera de combustió. Per tal de dur-ho a terme s'ha desenvolupat una metodología numèrica basada en CFD capaç de modelar el flux bifàsic no reactiu a l'interior d'un cremador LDI acadèmic mitjançant els enfocaments de turbulència U-RANS i LES en un marc Eulerià-Lagrangià. La resolució numèrica d'aquest problema multiescala s'aborda mitjançant la resolució completa del flux al llarg de tots els elements que constitueixen la maqueta experimental, incloent el seu pas pel swirler i l'entrada a la càmera de combustió. Açò es duu a terme a través de dos codis CFD que involucren estratègies de mallat diferents: una basada en la generación automàtica de la malla i en l'algoritme de refinament adaptatiu (AMR) amb CONVERGE i l'altra que es basa en una tècnica de mallat estàtic més tradicional amb OpenFOAM. D'una banda, s'ha definit una metodologia per tal d'obtindre una estrategia de mallat òptima mitjançant l'ús de l'AMR i s'han explotat els seus beneficis front als enfocaments tradicionals de malla estàtica. D'aquesta forma, s'ha demostrat que l'aplicabilitat de les ferramente de control de malla disponibles en CONVERGE com el refinament fixe (fixed embedding) i l'AMR són una opció molt interessant per tal d'afrontar aquest tipus de problemes multiescala. Els resultats destaquen una optimització de l'ús dels recursos computacionals i una major precisió en les simulacions realitzades amb la metodologia presentada. D'altra banda, l'ús d'eines CFD s'ha combinat amb l'aplicació de tècniques de descomposició modal avançades (Proper Orthogonal Decomposition and Dynamic Mode Decomposition). La identificació numèrica dels principals modes acústics a la càmera de combustió ha demostrat el potencial d'aquestes ferramentes al permetre caracteritzar les estructures de flux coherents generades com a conseqüència del trencament dels vòrtex (VBB) i dels raigs fortament arremolinats presents al cremador LDI. A més, la implantació d'estos procediments matemàtics ha permès recuperar informació sobre les característiques de la dinàmica del flux i proporcionar un enfocament sistemàtic per tal d'identificar els principals mecanismes que sustenten les inestabilitats a la càmera de combustió. Finalment, la metodologia validada ha sigut explotada a traves d'un Diseny d'Experiments (DoE) per tal de quantificar la influència dels factors crítics de disseny en el flux no reactiu. D'aquesta manera, s'ha avaluat la contribución individual d'alguns paràmetres funcionals (el nombre de pales del swirler, l'angle de les pales, l'amplada de la càmera de combustió i la posició axial de l'orifici de l'injector) en els patrons del camp fluid, la distribució de la mida de gotes del combustible líquid i l'aparició d'inestabilitats en la càmera de combustió mitjançant una matriu ortogonal L9 de Taguchi. Aquest estudi estadístic és un bon punt de partida per a futurs estudis de injecció, atomització i combustió en cremadors LDI.
[EN] Aeronautical gas turbine engines present the main challenge of increasing the efficiency of the cycle while keeping the pollutant emissions below stringent restrictions. This has led to the design of new injection-combustion strategies working on more risky and problematic operating points such as those close to the lean extinction limit. In this context, the Lean Direct Injection (LDI) concept has emerged as a promising technology to reduce oxides of nitrogen (NOx) for next-generation aircraft power plants In this context, this thesis aims at contributing to the knowledge of the governing physical mechanisms within an LDI burner and to provide analysis tools for a deep characterisation of such complex flows. In order to do so, a numerical CFD methodology capable of reliably modelling the 2-phase nonreacting flow in an academic LDI burner has been developed in an Eulerian-Lagrangian framework, using the U-RANS and LES turbulence approaches. The LDI combustor taken as a reference to carry out the investigation is the laboratory-scale swirled-stabilised CORIA Spray Burner. The multi-scale problem is addressed by solving the complete inlet flow path through the swirl vanes and the combustor through two different CFD codes involving two different meshing strategies: an automatic mesh generation with adaptive mesh refinement (AMR) algorithm through CONVERGE and a more traditional static meshing technique in OpenFOAM. On the one hand, a methodology to obtain an optimal mesh strategy using AMR has been defined, and its benefits against traditional fixed mesh approaches have been exploited. In this way, the applicability of grid control tools available in CONVERGE such as fixed embedding and AMR has been demonstrated to be an interesting option to face this type of multi-scale problem. The results highlight an optimisation of the use of the computational resources and better accuracy in the simulations carried out with the presented methodology. On the other hand, the use of CFD tools has been combined with the application of systematic advanced modal decomposition techniques (i.e., Proper Orthogonal Decomposition and Dynamic Mode Decomposition). The numerical identification of the main acoustic modes in the chamber have proved their potential when studying the characteristics of the most powerful coherent flow structures of strongly swirled jets in a LDI burner undergoing vortex breakdown (VBB). Besides, the implementation of these mathematical procedures has allowed both retrieving information about the flow dynamics features and providing a systematic approach to identify the main mechanisms that sustain instabilities in the combustor. Last, this analysis has also allowed identifying some key features of swirl spray systems such as the complex pulsating, intermittent and cyclical spatial patterns related to the Precessing Vortex Core (PVC). Finally, the validated methodology is exploited through a Design of Experiments (DoE) to quantify the influence of critical design factors on the non-reacting flow. In this way, the individual contribution of some functional parameters (namely the number of swirler vanes, the swirler vane angle, the combustion chamber width and the axial position of the nozzle tip) into both the flow field pattern, the spray size distribution and the occurrence of instabilities in the combustion chamber are evaluated throughout a Taguchi's orthogonal array L9. Such a statistical study has supposed a good starting point for subsequent studies of injection, atomisation and combustion on LDI burners.
Belmar Gil, M. (2020). Computational study on the non-reacting flow in Lean Direct Injection gas turbine combustors through Eulerian-Lagrangian Large-Eddy Simulations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/159882
TESIS

High frequency combustion instabilities in liquid propellant rocket engines are spontaneously occurring pressure fluctuations that are coupled with unsteady combustion processes. Under the right conditions the unsteady fluctuations can grow to a point where they affect the operation of the combustion chamber. The cause of combustion instabilities, including which processes are responsible and under what conditions they arise, are not yet fully understood. The ability to predict and prevent combustion instabilities during the design of new combustion chambers, through better understanding, would dramatically reduce the uncertainty and risk in the development of new engines. An experimental combustor, designated BKH, is used to conduct high frequency combustion instability experiments. BKH operates with liquid oxygen and gaseous hydrogen propellants at supercritical conditions analogous to real rocket engines. The chamber features an acoustic excitation system that imposes an acoustic disturbance representative of a high frequency instability upon a cluster of five coaxial injection elements in the center of the chamber. The response of the elements to the imposed acoustic disturbance is observed using high speed optical diagnostics. The main aim of this project is to develop methods for predicting the flame response to high frequency acoustic forcing representative of combustion instability phenomena. BKH is employed as an experimental and numerical test case for investigating the flame response. Modelling and complementary data analysis methods are developed and applied to model the chamber flow field, identify and predict the excited acoustic disturbance, identify the flame response using optical data, and to predict the flame response numerically. The BKH experiments are first characterised by modelling the chamber numerically and determining the local acoustic disturbance acting upon the flame. A steady state chamber model with supercritical oxygen-hydrogen combustion was computed using a specialised CFD code. The model results indicate the secondary injection in BKH has a strong influence on the resulting flame distribution. A method for reconstructing the acoustic field from dynamic pressure sensor data was developed to determine the local acoustic disturbance acting upon the combustion zone over a range of excitation frequencies. A low-order acoustic modelling approach is also shown to predict the resonant mode frequencies and the evolution of the acoustic field. The flame response to the imposed acoustic disturbance is identified by analysing optical data from BKH experiments and unsteady CFD modelling. Multi-variable dynamic mode decomposition (DMD) analysis is used to isolate the flame response to the imposed acoustic disturbance in shadowgraph and OH* imaging data. Wave-like structures propagating along the surface of the liquid oxygen (LOx) jet and a phase difference of 45° between acoustic pressure and observed intensity fluctuations were identified. An unsteady model of an injection element subjected to representative acoustic forcing is used to predict the flame response for a range of excitation amplitudes. Velocity ratio fluctuations caused by acoustic coupling with the oxidiser post in a pressure antinode are identified. The trend of exponential decay of the length of the LOx core with increasing transverse acoustic amplitude excitation is reproduced numerically and the flattening and flapping motion of the flame was further investigated using the numerical results.
Thesis (Ph.D.) -- University of Adelaide, School of Mechanical Engineering, 2017.

A research-scale solid-fuel ramjet test article has been developed to study the secondary combustion zone of solid fuel ramjets. Tests were performed at a constant core air mass flowrate of 0.77 kg/s with 0%, 15%, and 30% bypass ratios. The propulsive performance analysis results indicate that the 0% bypass case had the highest regression rate and fuel mass flowrate. The regression rate and fuel mass flowrate of fuel without carbon black was the lowest. The specific impulse with air mass flowrate included was highest for the 0% bypass case reaching 130 s and lowest for the 30% bypass case reaching 110 s. For specific impulse with air mass flowrate excluded, the 30% bypass case achieved 2,800 s while the 0% bypass case achieved 1,800 s. The characteristic velocity was greatest for 0% bypass reaching 1,025 m/s and lowest for 30% bypass reaching 900 m/s. The combustion efficiency was highest for the 15% bypass case with carbon black addition approaching 0.82. 50 kHz and 75 kHz CH* chemiluminescence imaging was performed. Analyzing thin slivers of the images over 40,001 frames with frequency-domain techniques showed that most of the high amplitude content occurred below 1-5kHz with small peaks near 20 kHz and 30 kHz. Dynamic mode decomposition (DMD) was performed on sets of 10,001 spatially-calibrated images and their corresponding uncalibrated, uncropped images. Most of the tests exhibited low-frequency axial pumping, transverse modes, and other mode shapes indicative of the secondary injection. The prominence of transverse and other jet-related modes over axial modes appeared to be related to increasing bypass ratio. High-frequency axial modes also appeared in a case thought to have high core-flow momentum that did not appear at these high frequencies for other cases. The DMD modes for 0% bypass were indiscernible due to high soot content. Most of the modes corresponding to the calibrated images also appeared in the uncalibrated images, however, with different mode amplitude rankings. PIV was performed at 5 kHz for one test at 15% bypass. The instantaneous vector fields for these tests displayed local velocities up to 600 m/s. The mean images showed velocities up to 250 m/s. The two-dimensional turbulent kinetic energies reached 200 m2/s2 in several regions throughout the flowfield. The turbulence intensity exceeded 0.20 near the bottom of the flowfield.

Detonation is a supersonic mode of combustion that is modeled by a system of conservation laws of compressible fluid mechanics coupled with the equations describing thermodynamic and chemical properties of the fluid. Mathematically, these governing equations admit steady-state travelling-wave solutions consisting of a leading shock wave followed by a reaction zone. However, such solutions are often unstable to perturbations and rarely observed in laboratory experiments. The goal of this work is to study the stability of travelling-wave solutions of detonation models by the following novel approach. We linearize the governing equations about a base travelling-wave solution and solve the resultant linearized problem using high-order numerical methods. The results of these computations are postprocessed using dynamic mode decomposition to extract growth rates and frequencies of the perturbations and predict stability of travelling-wave solutions to infinitesimal perturbations. We apply this approach to two models based on the reactive Euler equations for perfect gases. For the first model with a one-step reaction mechanism, we find agreement of our results with the results of normal-mode analysis. For the second model with a two-step mechanism, we find that both types of admissible travelling-wave solutions exhibit the same stability spectra. Then we investigate the Fickett’s detonation analogue coupled with a particular reaction-rate expression. In addition to the linear stability analysis of this model, we demonstrate that it exhibits rich nonlinear dynamics with multiple bifurcations and chaotic behavior.

Combustion instability, a complex phenomenon observed in combustion chambers is due to the coupling between heat release and other unsteady flow processes. Combustion instability has long been a topic of interest to rocket scientists and has been extensively investigated experimentally and computationally. However, to date, there is no computational tool that can accurately predict the combustion instabilities in full-size combustors because of the amount of computational power required to perform a high-fidelity simulation of a multi-element chamber. Hence, the focus is shifted to reduced fidelity computational tools which may accurately predict the instability by using the information available from the high-fidelity simulations or experiments of single or few-element combustors. One way of developing reduced fidelity computational tools involves using a reduced fidelity solver together with the flame transfer functions that carry important information about the flame behavior from a high-fidelity simulation or experiment to a reduced fidelity simulation.

To date, research has been focused mainly on premixed flames and using acoustic solvers together with the global flame transfer functions that were obtained by integrating over a region. However, in the case of rockets, the flame is non-premixed and distributed in space and time. Further, the mixing of propellants is impacted by the level of flow fluctuations and can lead to non-uniform mean properties and hence, there is a need for reduced fidelity solver that can capture the gas dynamics, nonlinearities and steep-fronted waves accurately. Nonlinear Euler equations have all the required capabilities and are at the bottom of the list in terms of the computational cost among the solvers that can solve for mean flow and allow multi-dimensional modeling of combustion instabilities. Hence, in the current work, nonlinear Euler solver together with the spatially distributed local flame transfer functions that capture the coupling between flame, acoustics, and hydrodynamics is explored.

In this thesis, the approach to extract flame transfer functions from high-fidelity simulations and their integration with nonlinear Euler solver is presented. The dynamic mode decomposition (DMD) was used to extract spatially distributed flame transfer function (FTF) from high fidelity simulation of a single element non-premixed flame. Once extracted, the FTF was integrated with nonlinear Euler equations as a fluctuating source term of the energy equation. The time-averaged species destruction rates from the high-fidelity simulation were used as the mean source terms of the species equations. Following a variable gain approach, the local species destruction rates were modified to account for local cell constituents and maintain correct mean conditions at every time step of the nonlinear Euler simulation. The proposed reduced fidelity model was verified using a Rijke tube test case and to further assess the capabilities of the proposed model it was applied to a single element model rocket combustor, the Continuously Variable Resonance Combustor (CVRC), that exhibited self-excited combustion instabilities that are on the order of 10% of the mean pressure. The results showed that the proposed model could reproduce the unsteady behavior of the CVRC predicted by the high-fidelity simulation reasonably well. The effects of control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size are also studied. The reduced fidelity model could reproduce the limit cycle amplitude within a few percent of the mean pressure. The successful constraints on the model include good spatial resolution and FTF with all modes up to at least one dominant frequency higher than the frequencies of interest. Furthermore, the reduced fidelity model reproduced consistent mode shapes and linear growth rates that reasonably matched the experimental observations, although the apparent ability to match growth rates needs to be better understood. However, the presence of significant heat release near a pressure node of a higher harmonic mode was found to be an issue. This issue was rectified by expanding the pressure node of the higher frequency mode. Analysis of two-dimensional effects and coupling between the local pressure and heat release fluctuations showed that it may be necessary to use two dimensional spatially distributed local FTFs for accurate prediction of combustion instabilities in high energy devices such as rocket combustors. Hybrid RANS/LES-FTF simulation of the CVRC revealed that it might be necessary to use Flame Describing Function (FDF) to capture the growth of pressure fluctuations to limit cycle when Navier-Stokes solver is used.

The main objectives of this thesis are:

1. Extraction of spatially distributed local flame transfer function from the high fidelity simulation using dynamic mode decomposition and its integration with nonlinear Euler solver

2. Verification of the proposed approach and its application to the Continuously Variable Resonance Combustor (CVRC).

3. Sensitivity analysis of the reduced fidelity model to control parameters such as the number of modes included in the FTF, the number of sampling points used in the Fourier transform of the unsteady heat release, and mesh size.

The goal of this thesis is to contribute towards a reduced fidelity computational tool which can accurately predict the combustion instabilities in practical systems using flame transfer functions, by providing a path way for reduced fidelity multi-element simulation, and by defining the limitations associated with using flame transfer functions and nonlinear Euler equations for non-premixed flames.