We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results. / We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results.

This work is concerned with the introduction of a new higher order numerical scheme based on the discontinuous Galerkin method (DGM). We follow the methodology of higher order finite volume (HOFV) and spectral volume (SV) schemes and introduce a reconstruction operator into the DGM. This operator constructs higher order piecewise polynomial reconstructions from the lower order DGM scheme. We present two variants: the generalization of standard HOFV schemes, already proposed by Dumbser et al. (2008) and the generalization of the SV method introduced by Wang (2002). Theoretical aspects are discussed and numerical experiments with the focus on a 2D advection problem are carried out. Powered by TCPDF (www.tcpdf.org)

Les méthodes classiques de résolution des équations de l'élasticité linéaire sont les méthodes éléments finis. Ces méthodes produisent de très bons résultats et sont très largement analysées mathématiquement pour l'étude des déformations solides. Pour des problèmes de couplage solide/fluide, pour des situations réalistes en présence de discontinuités (modélisation des fronts de gel dans les sols humides), ou bien encore pour des domaines de calcul mieux adaptés aux maillages non conformes, il parait intéressant de disposer de solveurs Volumes Finis. Les méthodes Volumes Finis sont très largement utilisées en mécanique des fluides. Appliquées aux problèmes de convection, elles sont bien adaptées à la capture de solutions présentant des discontinuités et ne nécessitent pas de maillages conformes. De plus, elles présentent l'avantage de conserver au niveau discret les flux à travers les interfaces du maillage. C'est pourquoi sont développées et testées, dans cette thèse, plusieurs méthodes de volumes finis, qui permettent de traiter le problème de l'élasticité. On a, dans un premier temps, mis en œuvre la méthode LSGR (Least Squares Gradient Reconstruction), qui reconstruit des gradients par volumes à partir d'une formule de moindres carrés pondérés sur les volumes voisins. Elle est testée pour des maillages tétraédriques non structurés, et montre un ordre 1 de convergence. La méthode des Volumes Finis mixtes est ensuite présentée, basée sur la conservation d'un flux "pénalisé" à travers les interfaces. Cette pénalisation impose une contrainte sur le type de maillage utilisé, et des tests sont réalisés en 2d avec des maillages structurés et non structurés de quadrangles. On étend ensuite la méthode des Volumes Finis diamants à l'élasticité. Cette méthode détermine un gradient discret sur des sous volumes associés aux interfaces à partir de l'interpolation de la solution aux sommets du maillage. La convergence théorique est prouvée sous réserve de vérifier une condition de coercivité. Les résultats numériques, en 2d pour des maillages non structurés, conduisent à un ordre de convergence meilleur que celui prouvé. Enfin, la méthode DDFV (Discrete Duality Finite Volume), qui est une extension de la méthode Diamant, est présentée. Elle est basée sur une correspondance entre plusieurs maillages afin d'y construire des opérateurs discrets en "dualité discrète". On montre que la méthode est convergente d'ordre 1. Les illustrations numériques, réalisées en 2d et en 3d pour des maillages non structurés, montrent une convergence d'ordre 2, ce qui est fréquemment observé pour cette méthode. / Finite element methods are conventionally used for solving linear elasticity equations. These methods produce very good results and are widely analyzed from a mathematical point of view to study solid deformations. It seems interesting to have Finite Volume solvers for coupled solid/fluid problems, realistic situations in presence of discontinuities (freezing fronts modeling in wet soils), or even to compute fields better suited to non-conforming meshes. Finite Volume methods are widely used in fluid mechanics. Applied to convection problems, they are well suited to compute solutions with discontinuities and do not require mesh conformity. Moreover, they have the advantage of preserving discrete flows across the interfaces of the mesh. Therefore, we develop and test in this thesis several finite volume methods for solving the elasticity problem. First of all, we implement the LSGR method (Least Squares Gradient Reconstruction), which reconstructs gradients by volume from a weighted least squares formula on neighboring volumes. This method has been successfully tested for unstructured tetrahedral meshes, and shows a first-order convergence rate. Then, we present the Mixed Finite Volume method, based on the conservation of a "penalized" flow across the interfaces. The penalty term imposes a constraint on the type of meshes, and numerical tests are performed in 2D with structured and unstructured quadrangles. Afterwards, we extend the diamond-cell Finite Volume method to the elasticity. This method computes a discrete gradient on sub-volumes related to the interfaces from the interpolation of the solution at vertices. The theoretical convergence is proved under a coercivity condition. The numerical results, achieved in 2d for unstructured meshes, give a second-order convergence rate. Finally, we present the DDFV method (Discrete Duality Finite Volume), which is an extension of the precedent one. This method is based on a correspondence between several meshes in order to construct discrete operators on "discrete duality". We show that the DDFV scheme is a first-order convergent method. The 2d and 3d numerical tests on unstructured meshes show a second-order convergence rate, which is a classical result for this method.

Dans certaines conditions de température et de débit, l’eau naturellement présente dans le kérosène va givrer l’intérieur des conduites du système carburant avion. Ces dépôts peuvent libérer des particules de glace qui sont entrainées par l’écoulement, et provoquent le colmatage des équipements hydrauliques situés en aval. Ce phénomène fut mis en évidence suite à l’accident d’un Boeing 777 en 2008, aussi sa compréhension est un enjeu important pour les acteurs de l’industrie aéronautique. Un dispositif a été spécialement conçu pour reproduire cette menace de façon quantifiée. De l’eau est atomisée dans un écoulement à basse température, puis cristallise pour former une suspension qui vient colmater différentes cibles perforées. Les températures, débits et pertes de charge sont mesurées, et le phénomène est filmé par une caméra haute fréquence. Un modèle a été réalisé à partir de cesobservations, complétées par des données issues de la littérature et de retoursd’expérience. Pour la phase fluide, les équations de Navier-Stokes incompressibles sont résolues par une approche volumes finis. Le couplage pression-vitesse est obtenu par l’algorithme SIMPLE et l’ordre élevé au moyen de la méthode MLS. La phase solide est simulée par éléments discrets. L’interaction fluide-particules repose sur une approche de type milieu poreux. Un code CFD-DEM parallèle a été développé, et les premières simulations d’écoulement en milieu granulaire sont en bon agrément avec des résultats expérimentaux. / Water, which exists naturally in jet-engine fuel, may freeze within theaircraft fuel pipes under certain temperatures and flow rates. The ice particles released by these deposits are entrained by the flow, and clog the hydraulics downstream. The understanding of this phenomenon, highlighted by the crash of a Boeing 777 in 2008, is an important issue for the aviation industry. Therefore a device has been designed to reproduce this threat in a controlled and quantified way. Water is atomized in low temperature jet-engine fuel and the droplets crystallize. The resulting slurry clogs different kinds of perforated targets. Temperatures, flow rates and pressure drops are monitored, and the phenomenon is filmed by a high frequency camera. A model was constructed based on these observations and data from literature and feedbacks. For the fluid phase, the incompressible Navier-Stokes equations are solved within a finite volume framework. The pressure-velocity coupling is achieved using the SIMPLE algorithm and high order of accuracy thanks to the MLS method. The solid phase is simulated using discrete elements. The fluid-particle interaction is based on a porous medium approach. A CFD-DEM parallel code has been developed to run the model. The first simulations of flow through granular media are in good agreement with experimental results.

Este trabalho discute a questão da interação entre o casco do navio e o propulsor em funcionamento conjunto, e sua simulação por ferramentas computacionais. O texto se concentra em descrever os principais efeitos dessa interação, as principais dificuldades em se estimar esses efeitos, os métodos tradicionalmente usados, e como ferramentas computacionais podem ser aplicadas de maneira vantajosa. No texto também pode ser encontrada uma análise crítica dos métodos mais comuns e dos resultados, baseada em trabalhos de diversos autores, publicados nacional e internacionalmente. É apresentado o método dos volumes finitos, usado nesta pesquisa, algumas de suas particularidades principais, vantagens e desvantagens, e os resultados das simulações realizadas, interpretados à luz dos valores experimentais usados para comparação e das limitações do método dos Volumes Finitos. A comparação é feita analisando-se grandezas integrais, como a resistência do casco ou o empuxo do propulsor; e também as características do escoamento, como o perfil de velocidade na esteira, presença de vórtices, e outras estruturas típicas. Os resultados obtidos têm a mesma ordem de precisão dos que tem sido obtidos por outros pesquisadores, internacionalmente, e são coerentes qualitativamente; mas algumas questões referentes aos modelos físico e numérico escolhidos ainda limitam a precisão dos resultados e restringem sua adoção em atividades de engenharia. Entretanto, diversas características observadas no escoamento contribuem para aumentar o conhecimento de alguns fenômenos envolvidos no problema. / This work is about the hull and propeller interaction in joint functioning, and its simulation by computational tools. The text concentrates in describing the main effects of such interaction, the main difficulties in the estimation of these effects, the methods traditionally adopted, and how computational tools can be applied in advantageous way. A critical analysis of the most common methods and results, based on paperworks of several different authors worldwide, can also be found. The Finite Volumes method, used in this research, is presented - its main issues, advantages and disadvantages, and the simulations outcomes, compared to the experimental values and explained by the knowledge of the Finite Volumes method limitations. The comparison is made by means of both integral values, such as the hull\'s resistance or the propeller thrust; as well as the characteristics of the flow, like the wake velocity profile, presence of vortex, and other typical structures. The results shows the same error band than the ones which has been obtained by other researchers, worldwide, and most of the typical characteristics of the flow were observed. But some issues concerned to the chosen physical and numerical models still limit the precision of the outcomes, and restrict the application of such models at engineering activities. But several insights about the flow, obtained at this study, can be useful to the understanding of some phenomena involved in the propeller operation.

The objective of the present work is the analysis of coextrusion processes by numerical simulation based on phase-field modeling of stratified confined flows. The study of such flows is motivated by the presence of complex phenomena appearing in a vast range of industrial operational coextrusion conditions due to the differences in the components properties and their viscoelastic behavior. The basic idea in coextrusion is to combine several layers of different polymers in a common die, to form a unique product with enhanced properties. However, the existence of fluid stratification in the die is responsible of a severe distortion of the interface between the fluid components, causing a loss of efficiency for the whole process. Experimental data show that, even if a stratified initial configuration is imposed at the die entry, one fluid eventually encapsulates the other in most of the flow condition analyzed. The intrinsically three-dimensional nature of this phenomenon has required the development of a three-dimensional flow solver based on the finite volume discretization of the Navier-Stokes equations for incompressible and isothermal flow, together with differential nonlinear constitutive equations (Giesekus, PTT models). The presence of two fluid phases is taken into account by a phase field model that implies the solution of an additional scalar equation to describe the evolution of the interface on a fixed Eulerian grid. This model, unlike others of the same family, has a thermodynamic derivation and can be physically interpreted. The proposed method is tested against experimental data and solutions already available in literature and a study of coextrusion in rectangular dies is performed to identify the dependence of encapsulation on the flow parameters

Este trabalho discute a questão da interação entre o casco do navio e o propulsor em funcionamento conjunto, e sua simulação por ferramentas computacionais. O texto se concentra em descrever os principais efeitos dessa interação, as principais dificuldades em se estimar esses efeitos, os métodos tradicionalmente usados, e como ferramentas computacionais podem ser aplicadas de maneira vantajosa. No texto também pode ser encontrada uma análise crítica dos métodos mais comuns e dos resultados, baseada em trabalhos de diversos autores, publicados nacional e internacionalmente. É apresentado o método dos volumes finitos, usado nesta pesquisa, algumas de suas particularidades principais, vantagens e desvantagens, e os resultados das simulações realizadas, interpretados à luz dos valores experimentais usados para comparação e das limitações do método dos Volumes Finitos. A comparação é feita analisando-se grandezas integrais, como a resistência do casco ou o empuxo do propulsor; e também as características do escoamento, como o perfil de velocidade na esteira, presença de vórtices, e outras estruturas típicas. Os resultados obtidos têm a mesma ordem de precisão dos que tem sido obtidos por outros pesquisadores, internacionalmente, e são coerentes qualitativamente; mas algumas questões referentes aos modelos físico e numérico escolhidos ainda limitam a precisão dos resultados e restringem sua adoção em atividades de engenharia. Entretanto, diversas características observadas no escoamento contribuem para aumentar o conhecimento de alguns fenômenos envolvidos no problema. / This work is about the hull and propeller interaction in joint functioning, and its simulation by computational tools. The text concentrates in describing the main effects of such interaction, the main difficulties in the estimation of these effects, the methods traditionally adopted, and how computational tools can be applied in advantageous way. A critical analysis of the most common methods and results, based on paperworks of several different authors worldwide, can also be found. The Finite Volumes method, used in this research, is presented - its main issues, advantages and disadvantages, and the simulations outcomes, compared to the experimental values and explained by the knowledge of the Finite Volumes method limitations. The comparison is made by means of both integral values, such as the hull\'s resistance or the propeller thrust; as well as the characteristics of the flow, like the wake velocity profile, presence of vortex, and other typical structures. The results shows the same error band than the ones which has been obtained by other researchers, worldwide, and most of the typical characteristics of the flow were observed. But some issues concerned to the chosen physical and numerical models still limit the precision of the outcomes, and restrict the application of such models at engineering activities. But several insights about the flow, obtained at this study, can be useful to the understanding of some phenomena involved in the propeller operation.

Retracer l’histoire d’un bassin est un préalable essentiel à toute recherche d’hydrocarbures. Pour cela, on a recours à un modèle stratigraphique, qui simule l'évolution des bassins sédimentaires sur de grandes échelles de temps (millions d'années) et d'espace (centaines de kilomètres). Le logiciel Dionisos, développé à IFPEN depuis 1992 et très apprécié par les compagnies pétrolières, permet d’effectuer ce type de calculs en prenant en compte deux grands processus physiques : (1) le transport gravitaire des sédiments dû à l’inclinaison du sol ; (2) l’écoulement de l’eau provenant des fleuves et des précipitations. Le transport gravitaire est décrit par une équation de diffusion dans laquelle le flux de sédiments dépend de la pente du sol. Initialement, cette dépendance est linéaire. Pour mieux s’approcher des observations réelles, on souhaite la rendre non-linéaire par l’intermédiaire d’un p-Laplacien. Ce changement nécessite la conception d’une nouvelle méthode de résolution numérique, qui doit offrir non seulement une grande rapidité d’exécution, mais aussi des garanties de robustesse et de précision des résultats. De plus, elle doit être compatible avec une contrainte sur le taux d’érosion présente dans le modèle. L’ajout de l’écoulement de l’eau est aussi une sophistication récente du modèle physique de Dionisos. Il se traduit par l’introduction d’une nouvelle équation aux dérivées partielles, couplée à celle du transport. Là encore, il est important d’élaborer une stratégie de résolution numérique innovante, en ce sens qu’elle doit être à la fois performante et bien adaptée au fort couplage de ces deux phénomènes. L'objectif de cette thèse est de moderniser le cœur numérique de Dionisos afin de traiter plus adéquatement les processus physiques ci-dessus. On cherche notamment à élaborer un schéma implicite par rapport à toutes les inconnues qui étend et améliore le schéma actuel. Les méthodologies retenues serviront de base à la prochaine génération du calculateur. / An essential prerequisite to finding hydrocarbons is to trace back the history of a basin. To this end, geologists resort to a stratigraphic model, which simulates the evolution of sedimentary basins over large time scales (million years) and space (hundreds of kilometers). The Dionisos software, developed by IFPEN since 1992 and highly praised by oil companies, makes this type of calculation possible by accounting for two main physical processes: (1) the sediment transport due to gravity; (2) the flow of water from rivers and rains. The gravity transport is described by a diffusion equation in which the sediment flow depends on the slope of the ground. Initially, this dependence is linear. To better match experimental observations, we wish to make it nonlinear by means of a p-Laplacian. This upgrade requires to design a dedicated numerical method which should not only run fast but also provide guarantees of robustness and accuracy. In addition, it must be compatible with a constraint on the erosion rate in the present model. The water flow due to rivers and rains is also a recent enhancement brought to the physical model of Dionisos. This is achieved by introducing a new partial differential equation, coupled with that of sediment transport. Again, it is capital to work out an innovative numerical strategy, in the sense that it must be both efficient and well suited to the strong coupling of these two phenomena. The objective of this thesis is to rejuvenate the numerical schemes that lie at the heart of Dionisos in order to deal more adequately with the physical processes above. In particular, we look for an implicit scheme with respect to all the unknowns that extends and improves the current scheme. The methodologies investigated in this work will serve as a basis for the next generation of stratigraphic modelling softwares.

L'ensemble des travaux présentés dans cette thèse porte sur la simulation numérique des procédés de coextrusion par un modèle d'écoulement stratifié basé sur la méthode du champ de phase. L'avantage technologique offert par la coextrusion réside dans la possibilité de combiner des matériaux ayant des propriétés physiques très spécifiques dans un produit unique. Toutefois, les différences rhéologiques entre les divers matériaux sont elles-mêmes responsables d'un phénomène de distorsion de l'interface séparant deux couches adjacents. Les données expérimentales en coextrusion bicouches montrent que, en raison des différences de viscosité et d'élasticité entre le deux composants, le fluide le moins visqueux encapsule le fluide plus visqueux et le passage d'une configuration stratifiée à une encapsulée comporte une perte de qualité du produit final. Ce phénomène, dit d'enrobage représente donc un sujet de très grande actualité dans la recherche industrielle et la compréhension des mécanismes le générant sera utile pour l'amélioration des procédés de mise en forme des polymères. La nature intrinsèquement tridimensionnelle de l'enrobage a requis le développement d'un code pour la simulation tridimensionnelle basée sur la méthode des volumes finis pour la discrétisation des équations de Navier-Stokes pour les écoulement incompressibles et isothermes couplées avec une loi constitutive différentielle non linéaire (modèles de Giesekus ou PTT). La présence de deux fluides est prise en compte par une équation scalaire supplémentaire décrivant l'évolution de l'interface sur un maillage fixe. Cette équation offre une interprétation physique précise car elle est dérivée de la thermodynamique de séparation de phase d'un fluide binaire. Le modèle proposé est validé par confrontation avec les résultats expérimentaux et numériques disponibles dans la littérature. Une étude numérique de la coextrusion en filière rectangulaire est effectuée afin de mettre en évidence les facteurs influençant l'enrobage et la nature de son origine / The objective of the present work is the analysis of coextrusion processes by numerical simulation based on phase-field modeling of stratified confined flows. The study of such flows is motivated by the presence of complex phenomena appearing in a vast range of industrial operational coextrusion conditions due to the differences in the components properties and their viscoelastic behavior. The basic idea in coextrusion is to combine several layers of different polymers in a common die, to form a unique product with enhanced properties. However, the existence of fluid stratification in the die is responsible of a severe distortion of the interface between the fluid components, causing a loss of efficiency for the whole process. Experimental data show that, even if a stratified initial configuration is imposed at the die entry, one fluid eventually encapsulates the other in most of the flow condition analyzed. The intrinsically three-dimensional nature of this phenomenon has required the development of a three-dimensional flow solver based on the finite volume discretization of the Navier-Stokes equations for incompressible and isothermal flow, together with differential nonlinear constitutive equations (Giesekus, PTT models). The presence of two fluid phases is taken into account by a phase field model that implies the solution of an additional scalar equation to describe the evolution of the interface on a fixed Eulerian grid. This model, unlike others of the same family, has a thermodynamic derivation and can be physically interpreted. The proposed method is tested against experimental data and solutions already available in literature and a study of coextrusion in rectangular dies is performed to identify the dependence of encapsulation on the flow parameters

In the context of gradient-based numerical optimisation, the adjoint method is an e cient way of computing the gradient of the cost function at a computational cost independent of the number of design parameters, which makes it a captivating option for industrial CFD applications involving costly primal solves. The method is however a ected by instabilities, some of which are inherited from the primal solver, notably if the latter does not fully converge. The present work is an attempt at curbing primal solver limitations with the goal of indirectly alleviating adjoint robustness issues. To that end, a novel discretisation scheme for the steady-state incompressible Navier- Stokes problem is proposed: Mixed Hybrid Finite Volumes (MHFV). The scheme draws inspiration from the family of Mimetic Finite Di erences and Mixed Virtual Elements strategies, rid of some limitations and numerical artefacts typical of classical Finite Volumes which may hinder convergence properties. Derivation of MHFV operators is illustrated and each scheme is validated via manufactured solutions: rst for pure anisotropic di usion problems, then convection-di usion-reaction and nally Navier-Stokes. Traditional and novel Navier-Stokes solution algorithms are also investigated, adapted to MHFV and compared in terms of performance. The attention is then turned to the discrete adjoint Navier-Stokes system, which is assembled in an automated way following the principles of Equational Di erentiation, i.e. the di erentiation of the primal discrete equations themselves rather than the algorithm used to solve them. Practical/computational aspects of the assembly are discussed, then the adjoint gradient is validated and a few solution algorithms for the MHFV adjoint Navier-Stokes are proposed and tested. Finally, two examples of full shape optimisation procedures on internal ow test cases (S-bend and U-bend) are reported.