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This dissertation is concerned with the mathematical modeling of the flow in a porous cylinder with a focus on applications to solid rocket motors. After discussing the historical development and major contributions to the understanding of wall injected flows, we present an inviscid rotational model for solid and hybrid rockets with arbitrary headwall injection. Then, we address the problem of pressure integration and find that for a given divergence free velocity field, unless the vorticity transport equation is identically satisfied, one cannot find an analytic expression for the pressure by direct integration of the Navier-Stokes equations. This is followed by the application of a variational procedure to seek novel solutions with varying levels of kinetic energies. These are found to cover a wide spectrum of admissible motions ranging from purely irrotational to highly rotational fields. Subsequently, a second law analysis as well as an extension of Kelvin's energy theorem to open boundaries are presented to verify and corroborate the variational model. Finally, the focus is shifted to address the problem of laminar viscous flow in a porous cylinder with regressing walls. This is tackled using two different analytical techniques, namely, perturbation and decomposition. Comparisons with numerical Runge--Kutta solutions are also provided for a variety of wall Reynolds numbers and wall regression speeds.

Les oscillations de pression sont un enjeu majeur dans le design des moteurs à propergol solide car de faibles oscillations de pression (ODP) dans la chambre entraînent de fortes oscillations de poussée ce qui conduit à des vibrations néfastes pour les structures et les satellites embarqués. Les ODP sont encore aujourd'hui un vaste sujet de recherche et la simulation numérique est un outil indispensable dans leur analyse. De nombreux travaux ont permis de mettre en évidence divers mécanismes générateurs d'oscillations, mais la conception des nouveaux moteurs favorise la formation d'une instabilité hydrodynamique, appelée VSA et caractérisée par des détachements tourbillonnaire au niveau des discontinuités de la surface débitante. Etudiée dans les travaux sur le C1x [Vuillot 1995, Dupays 1996], il reste cependant divers points à aborder afin d'avoir une vision complète des mécanismes qui pilotent et modifient cette instabilité. Pour cela, il a été choisi dans ces travaux d'isoler le VSA dans une configuration académique et d'étudier dans un premier temps, l'impact du soufflage latéral, généré par un dégagement gazeux du à la combustion d'un bloc de propergol en aval de l'angle. Les deux approches utilisées, à savoir la simulation numérique et la stabilité linéaire, démontrent que le soufflage latéral possède un fort effet stabilisant sur le VSA. Dans un deuxième temps, l'impact de la combustion des particules d'aluminium et des résidus, présents dans un moteur à propergol solide, est analysé. Ces travaux montrent que les particules, via des mécanismes complexes, peuvent jouer à la fois un rôle stabilisant et déstabilisant sur le VSA. Pour finir, l'impact de la mise à l'échelle sur l'instabilité est étudié. Si en gaz seul, les résultats obtenus à échelle réduite sont directement transposables vers l'échelle réelle, la mise à l'échelle modifie le comportement des particules dans les structures tourbillonnaires et donc leur rôle sur l'instabilité. / Pressure oscillations (ODP) are a major issue in solid rocket motor design, as very small pressure oscillations induce strong thrust oscillations, involving vibrations detrimental to carrying load. ODP are still a vast and intense domain of research and the improvement of rocket motors mainly resorts to advanced numerical simulations. Extensive research have enabled to characterize several sources of instabilities, but new motor design promotes one hydrodynamic instability, called VSA and characterized by vortex shedding at geometry angles. VSA has be studied in the C1x configuration [Vuillot 1995, Dupays 1996] but several points still need to be studied in order to have a complete view of the phenomena driving and impacting this instability in a solid rocket motor flow. In this work, the VSA is isolated in an academic configuration and, in a first part, the impact of lateral blowing is studied. This blowing, never analysed so far, is due to burnt gases coming from the combustion of propellant block after the angle. This study has been performed following two approaches, numerical simulations and linear stability analysis. Both demonstrate the strong stabilizing effect of the lateral blowing. In a second part, the impact of aluminium particles combustion including the presence of residual particles, found in solid rocker motors, is analysed. This work shows that due to complex interaction mechanisms, particles can have a stabilizing or a destabilizing impact on the instability. Finally, the scaling impact is studied with and without particles. In purely gaseous configuration, the results obtained at reduced scale can be used directly at real scale as all the characteristics of the instability are preserved. However, with particles, the scaling modifies the particles behaviour and then the particles impact on the VSA.

This thesis presents the results and analyses of computational studies on certain problems of combustion instability in solid propellants. Specifically, effects of relaxing certain assumptions made in previous models of unsteady burning of solid propellants are investigated. Knowledge of unsteady burning of solid propellants is essential in studying the phenomenon of combustion instability in solid propellant rocket motors. In Chapter 1, an introduction to different types of unsteady combustion investigated in this thesis, such as 1) intrinsic instability, 2) pressure-driven dynamic burning, 3) extinction by depressurization, and 4) L* -instability, is given. Also, a review of previous experimental and theoretical studies of these phenomena is presented. From this review it is concluded that all the previous studies, which investigated the unsteady combustion of solid propellants, made one or more of the following assumptions: 1) quasi-steady gas-phase (QSG), 2) quasi-steady condensed phase reaction zone (QSC), 3) small perturbations, and 4) unity Lewis number. These assumptions limit the validity of the results obtained with such models to: 1) relatively low frequencies (< 1 kHz) of pressure oscillations and 2) small deviations in pressure from its steady state or mean values. The objectives of the present thesis are formulated based on the above conclusions. These are: 1) to develop a nonlinear numerical model of unsteady solid propellant combustion, 2) to relax the assumptions of QSG and QSC, 3) to study the consequent effects on the intrinsic instability and pressure-driven dynamic burning, and 4) to investigate the L* -instability in solid propellant rocket motors. In Chapter 2, a nonlinear numerical model, which relaxes the QSG and QSC assumptions, is set up. The transformation and nondimensionalization of the governing equations are presented. The numerical technique based on the method of operator-splitting, used to solve the governing equations is described. In Chapter 3, the effect of relaxing the QSG assumption on the intrinsic instability is investigated. The stable and unstable solutions are obtained for parameters corresponding to a typical composite propellant. The stability boundary, in terms of the nondimensional parameters identified by Denison and Baum (1961), is predicted using the present model. This is compared with the stability boundary obtained by previous linear stability theories, based on activation energy asymptotics in the gas-phase, which employed QSC and/or QSG assumptions. It is found that in the limit of large activation energy and low frequencies, present result approaches the previous theoretical results. This serves as a validation of the present method of solution. It is confirmed that relaxing the QSG assumption widens the stable region. However, it is found that a distributed reaction in the gas-phase destabilizes the burning. The effect of non-unity Lewis number on the stability boundary is also investigated. It is found that at parametric values corresponding to low pressures and large flame stand-off distances, small amplitude, high frequency (at frequencies near the characteristic frequency of the gas-phase) oscillations in burning rate appear when the Lewis number is greater than one. In Chapter 4, the effect of relaxing the QSG assumption is further investigated with respect to the pressure-driven dynamic burning. Comparison of the pressure-driven frequency response function, Rp, obtained with the present model, both in the QSG and non-QSG framework, with those obtained with previous linear stability theories invoking QSG and QSC assumptions are made. As the frequency of pressure oscillations approaches zero, |RP| predicted using present models approached the value obtained by previous theoretical studies. Also, it is confirmed that the effect of relaxing QSG is to decrease the |Rp| at frequencies near the first resonant frequency. Moreover, relaxing QSG assumption produces a second resonant peak in |Rp| at a frequency near the characteristic frequency of the gas-phase. Further, Rp calculated using the present model is compared with that obtained by a previous linear theory which relaxed the QSG assumption. The two models predicted the same resonant frequencies in the limit of small amplitudes of pressure oscillations. Finally, it is found that the effect of large amplitude of pressure oscillations is to introduce higher harmonics in the burning rate and to reduce the mean burning rate. In Chapter 5, first the present non-QSC model is validated by comparing its results with that of a previous non-QSC model for radiation-driven burning. The model is further validated for steady burning results by comparing with experimental data for a double base propellant (DBP). Then, the effect of relaxing the QSC assumption on steady state solution is investigated. It is found that, even in the presence of a strong gas-phase heat feedback, QSC assumption is valid for moderately large values of condensed phase Zel'dovich number, as far as steady state solution is concerned. However, for pressure-driven dynamic burning, relaxing the QSC assumption is found to increase |RP| at all frequencies. The error due to QSC assumption is found to become significant, either when |Rp| is large or as the driving frequency approaches the characteristic frequency of the condensed phase reaction zone. The predicted real part of the response function is quantitatively compared with experimental data for DBP. The comparison seems to be better with a value of condensed phase activation energy higher than that suggested by Zenin (1992). In Chapter 6, burning rate transients for a DBP during exponential depressurization are computed using non-QSG and non-QSC models. Salient features of extinction and combustion recovery are predicted. The predicted critical initial depressurization rate, (dp/dt)i, is found to decrease markedly when the QSC assumption is relaxed. The effect of initial pressure level on critical (dp/dt)i is studied. It is found that the critical (dp/dt)i decreases with the initial pressure. Also, the overshoot of burning rate during combustion recovery is found to be relatively large with low initial pressures. However as the initial pressure approached the final pressure, the reduction in initial pressure causes a large increase in the critical (dp/dt)i. No extinction is found to occur when the initial pressure is very close to the final pressure. In Chapter 7, a numerical model is developed to simulate the L* -instability in solid propellant motors. This model includes a) the propellant burning model that takes into account nonlinear pressure oscillations and that takes into account an unsteady gas- and condensed phase, and b) a combustor model that allows pressure and temperature oscillations of finite amplitude. Various regimes of L* -burning of a motor, with a typical composite propellant, namely 1) steady burning, 2) oscillatory burning leading to steady state, 3) oscillatory burning leading to extinction, 4) reignition and 5) chuffing are predicted. The predicted dependence of frequency of L* -oscillations on mean pressure is compared with one set of available experimental data. It is found that proper modeling of the radiation heat flux from the chamber walls to the burning surface may be important to predict the re-ignition. In Chapter 8, the main conclusions of the present study are summarized. Certain suggestions for possible future studies to enhance the understanding of dynamic combustion of solid propellants are also given.

Les gouttes d’oxyde d’aluminium présentes en masse dans l’écoulement interne des moteurs-fusées à propergol solide ont tendance à influerde façon importante sur l’écoulement et sur le fonctionnement du moteur quel que soit le régime. L’objectif de la thèse est d’améliorerles modèles diphasiques eulériens présents dans le code de calcul semi-industriel pour l’énergétique de l’ONERA, CEDRE, en y incluant lapossibilité d’une dispersion locale des particules en vitesse en plus de la dispersion en taille déjà présente dans le code, tout en gardant unestructure mathématique bien posée du système d’équations à résoudre. Cette nouvelle caractéristique rend le modèle capable de traiter lescroisements de trajectoires anisotropes, principale difficulté des modèles eulériens classiques pour les gouttes d’inertie modérément grande.En plus de la conception et de l’analyse détaillée d’une classe de modèles basés sur des méthodes de moments, le travail se concentre sur larésolution des systèmes d’équations obtenus en configurations industrielles. Pour cela, de nouvelles classes de schémas précis et réalisables pourle transport des particules dans l’espace physique et l’espace des phases sont développées. Ces schémas assurent la robustesse de la simulationmalgré différentes singularités (dont des chocs, -chocs, zones de pression nulle et zones de vide...) tout en gardant une convergence d’ordredeux pour les solutions régulières. Ces développements sont conduits en deux et trois dimensions, en plus d’un référentiel bidimensionnelaxisymétrique, dans le cadre de maillages non structurés.La capacité des schémas numériques à maintenir un niveau de précision élevé tout en restant robuste dans toutes les conditions est un pointclé pour les simulations industrielles de l’écoulement interne des moteurs à propergol solide. Pour illustrer cela, le code de recherche SIERRA,originellement conçu durant les année 90 pour les problématiques d’instabilités de fonctionnement en propulsion solide, a été réécrit afin depouvoir comparer deux générations de modèles et de méthodes numériques et servir de banc d’essais avant une intégration dans CEDRE. Lesrésultats obtenus confirment l’efficacité de la stratégie numérique choisie ainsi que le besoin d’introduire, pour les simulations axisymétriques,une condition à la limite spécifique, développée dans le cadre de cette thèse. En particulier, les effets à la fois du modèle et de la méthodenumérique dans le contexte d’une simulation de l’écoulement interne instationnaire dans les moteurs-fusées à propergol solide sont détaillés.Par cette approche, les liens entre des aspects fondamentaux de modélisation et de schémas numériques ainsi que leurs conséquences pour lesapplications sont mis en avant. / The massive amount of aluminum oxide particles carried in the internal flow of solid rocket motors significantly influences their behavior.The objective of this PhD thesis is to improve the two-phase flow Eulerian models available in the semi-industrial CFD code for energeticsCEDRE at ONERA by introducing the possibility of a local velocity dispersion in addition to the size dispersion already taken into accountin the code, while keeping the well-posed characteristics of the system of equations. Such a new feature enables the model to treat anisotropicparticle trajectory crossings, which is a key issue of Eulerian models for droplets of moderately large inertia.In addition to the design and detailed analysis of a class of models based on moment methods, the conducted work focuses on the resolution ofthe system of equations for industrial configurations. To do so, a new class of accurate and realizable numerical schemes for the transport ofthe particles in both the physical and the phase space is proposed. It ensures the robustness of the simulation despite the presence of varioussingularities (including shocks, -shocks, zero pressure area and vacuum...), while keeping a second order accuracy for regular solutions. Thesedevelopments are conducted in two and three dimensions, including the two dimensional axisymmetric framework, in the context of generalunstructured meshes.The ability of the numerical schemes to maintain a high level of accuracy in any condition is a key aspect in an industrial simulation of theinternal flow of solid rocket motors. In order to assess this, the in-house code SIERRA, originally designed at ONERA in the 90’s for solidrocket simulation purpose, has been rewritten, restructured and augmented in order to compare two generations of models and numericalschemes, to provide a basis for the integration of the features developed in CEDRE. The obtained results assess the efficiency of the chosennumerical strategy and confirm the need to introduce a new specific boundary condition in the context of axisymmetric simulations. Inparticular, it is shown that the model and numerical scheme can have an impact in the context of the simulation of the internal flow ofsolid rocket motors and their instabilities. Through our approach, the shed light on the links between fundamental aspects of modeling andnumerical schemes and their consequences on the applications.

Cette thèse vise à modéliser les instabilités hydrodynamiques générant des détachements tourbillonnaires pariétaux (ou VSP) responsables des Oscillations De Pression dans les moteurs à propergol solide longs et segmentés par interaction avec l’acoustique du moteur. Ces instabilités sont modélisées en tant que modes de stabilité linéaire globaux de l’écoulement d’un conduit à parois débitantes. En supposant que les structures pariétales émergent d’une perturbation de l’écoulement de base, des modes discrets et indépendants du maillage utilisé sont calculés. Dans ce but, une discrétisation par collocation spectrale multi-domaine est implémentée dans un solveur parallèle afin de s’affranchir de la croissance polynomiale des fonctions propres et de la présence de couches limites. Les valeurs propres ainsi calculées dépendent explicitement des frontières du domaine, à savoir la position de la perturbation et celle de la sortie, et sont ensuite validées par simulation numérique directe. On montre alors qu’elles permettent bien de décrire la réponse à une perturbation initiale de l’écoulement modifié par une rupture de débit pariétale. Ensuite, la simulation d’une réponse forcée par l’acoustique se fait sous forme de structures tourbillonnaires dont les fréquences discrètes sont en accord avec celles des modes de stabilité. Ces structures sont réfléchies en ondes de pression de même fréquences remontant l’écoulement. Finalement, la simulation numérique et la théorie de la stabilité permettent de montrer que le VSP, dont la réponse est linéaire vis-à-vis d’un forçage compressible comme l’acoustique, est le phénomène moteur des Oscillations De Pression. / The current work deals with the modeling of the hydrodynamic instabilities that play a major role in the triggering of the Pressure Oscillations occurring in large segmented solid rocket motors. These instabilities are responsible for the emergence of Parietal Vortex Shedding (PVS) and they interact with the boosters acoustics. They are first modeled as eigenmodes of the internal steady flowfield of a cylindrical duct with sidewall injection within the global linear stability theory framework. Assuming that the related parietal structures emerge from a baseflow disturbance, discrete meshindependant eigenmodes are computed. In this purpose, a multi-domain spectral collocation technique is implemented in a parallel solver to tackle numerical issues such as the eigenfunctions polynomial axial amplification and the existence of boundary layers. The resulting eigenvalues explicitly depend on the location of the boundaries, namely those of the baseflow disturbance and the duct exit, and are then validated by performing Direct Numerical Simulations. First, they successfully describe flow response to an initial disturbance with sidewall velocity injection break. Then, the simulated forced response to acoustics consists in vortical structures wihich discrete frequencies that are in good agreement with those of the eigenmodes. These structures are reflected into upstream pressure waves with identical frequencies. Finally, the PVS, which response to a compressible forcing such as the acoustic one is linear, is understood as the driving phenomenon of the Pressure Oscillations thanks to both numerical simulation and stability theory.