When traditional linearised theory is used to study free-surface flows past a surface-piercing object or over an obstruction in a stream, the geometry of the object is usually lost, having been assumed small in one or several of its dimensions. In order to preserve the nonlinear nature of the geometry, asymptotic expansions in the low-Froude or low-Bond limits can be derived, but here, the solution invariably predicts a waveless free-surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In this thesis, we will apply exponential asymptotics to the study of two new problems involving nonlinear geometries. In the first, we examine the case of free-surface flow over a step including the effects of both gravity and surface tension. Here, we shall see that the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects, leads to the discovery of a remarkable new set of solutions. In the second problem, we study the waves produced by bluff-bodied ships in low-Froude flows. We will derive the analytical form of the exponentially small waves for a wide range of hull geometries, including single-cornered and multi-cornered ships, and then provide comparisons with numerical computations. A particularly significant result is our confirmation of the thirty-year old conjecture by Vanden-Broeck & Tuck (1977) regarding the impossibility of waveless single-cornered ships.

Ce travail est consacré au développement de schémas numériques pour approcher les solutions de modèles d'écoulement type “depth averaged”. Dans un premier temps, nous détaillons la construction d'approches Volumes Finis pour le système Shallow Water avec termes sources sur maillages non structurés. En se basant sur une reformulation appropriée des équations, nous mettons en place un schéma équilibré et préservant la positivité de la hauteur d'eau, et suggérons des extensions MUSCL adaptées. La méthode est capable de gérer des topographies irrégulières et exhibe de fortes propriétés de stabilité. L'inclusion des termes de friction fait l'objet d'une analyse poussée, aboutissant à l'établissement d'une propriété type “Asymptotic Preserving” à travers l'amélioration d'un autre récent schéma Volumes Finis. La seconde composante de cette étude concerne les méthodes Elements Finis type Galerkin discontinu. Certaines des idées avancées dans le contexte Volumes Finis sont employées pour aborder le système Shallow Water surmaillages triangulaires. Des résultats numériques sont exposés et la méthode se révèle bien adaptée à la description d'une large variété d'écoulements. Partant de ces observations nous proposons finalement d'exploiter ces caractéristiques pour étendre l'approche à une nouvelle famille d'équations type Green-Nadghi. Des validations numériques sont également proposées pour valider le modèle numérique. / This work is devoted to the development of numerical schemes to approximatesolutions of depth averaged flow models. We first detail the construction of Finite Volume approaches for the Shallow Water system with source terms on unstructured meshes. Based on a suitable reformulation of the equations, we implement a well-balanced and positive preserving approach, and suggest adapted MUSCL extensions. The method is shown to handle irregular topography variations and demonstrates strong stabilities properties. The inclusion of friction terms is subject to a thorough analysis, leading to the establishment of some Asymptotic Preserving property through the enhancement of another recent Finite Volume scheme.The second aspect of this study concerns discontinuous Galerkin Finite Elementmethods. Some of the ideas advanced in the Finite Volume context areemployed to broach the Shallow Water system on triangular meshes. Numericalresults are exposed and the method turns out to be well suited to describe a large variety of flows. On these observations we finally propose to exploit its features to extend the approach to a new family of Green-Nadghi equations. Numerical experiments are also proposed to validate this numerical model.

No contexto do método MAC e baseado em esquemas de diferenças finitas, este trabalho apresenta três estudos: i) uma análise de estabilidade, ii) o desenvolvimento de técnicas implícitas e, iii) a construção de métodos de projeção para escoamentos com superfície livre. Na análise de estabilidade, o principal resultado mostra que o método de Crank-Nicolson torna-se condicionalmente estável quando aplicado para uma malha deslocada com a discretiza ção explícita das condições de contorno do tipo Dirichlet. Entretanto, o mesmo método com condições de contorno implícitas é incondicionalmente estável. Para obter métodos mais estáveis, formulações implícitas são desenvolvidas para a equação da pressão na superfície livre, derivada da condição de tensão normal. Esta estratégia resulta no acoplamento dos campos de velocidade e pressão, o que exige a introdução de novos métodos de projeção. Os métodos de projeção assim desenvolvidos resultam em novas metodologias para escoamentos com superfície livre que são apropriados para o tratamento de problemas com baixo número de Reynolds. Além disso, mostra-se que os métodos propostos podem ser aplicados para fluidos viscoelásticos. Novas estratégias são derivadas para obter métodos de projeção de segunda ordem de precisão para escoamentos com superfícies livres. Além dos resultados teóricos sobre a estabilidade de esquemas numéricos, técnicas implícitas e métodos de projeção, testes computacionais são realizados e comparados para consolidação da teoria apresentada. Os resultados numéricos são obtidos no sistema FREEFLOW. A eficiência e robustez das técnicas desenvolvidas neste trabalho são demonstradas na solução de problemas tridimensionais complexos com superfície livre e baixo número de Reynolds, incluindo os problemas do jato oscilante e do inchamento do extrudado / In the context of the MAC method and based on finite difference schemes, this work presents three studies: i) a stability analysis, ii) the development of implicit techniques, and iii) the construction of projection methods for free surface flows. In the stability analysis, the main result shows a precise stability restriction on the Crank-Nicolson method when one uses a staggered grid with Dirichlet explicit boundary conditions. However, the same method with implicit boundary conditions becomes unconditionally stable. In order to obtain more stable methods, implicit formulations are applied for the pressure equation at the free surface, which is derived from the normal stress condition. This approach results in a coupling of the velocity and pressure fields; hence new projection methods for free surface flows need to be developed. The developed projection methods result in new methodologies for low Reynolds number free surface flows. It is also shown that the proposed methods can be applied for viscoelastic fluids. New strategies are derived for obtaining second-order accurate projection methods for free surface flows. In addition to the theoretical results on the stability of numerical schemes, implicit techniques and projection methods, computational tests are carried out and the results compared to consolidate the theory. The numerical results are obtained by the FREEFLOW system. The eficiency and robustness of the techniques in this work are demonstrated by solving complex tridimensional problems involving free surface and low Reynolds numbers, including the jet buckling and the extrudate swell problems

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de problemas de interação fluido-estrutura (IFE) em aplicações envolvendo corpos rígidos submersos, onde escoamentos incompressíveis de fluidos Newtonianos com superfície livre são considerados. Propõe-se o emprego da Análise Isogeométrica por permitir a unificação entre os procedimentos de pré-processamento e análise, melhorando assim as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação da conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, em uma formulação cinemática ALE (Arbitrary Lagrangean- Eulerian). A consideração da superfície livre no escoamento se dá tratando o fluido como um meio bifásico, através do método Level Set. O corpo rígido apresenta não linearidade na rotação e restrições representadas por vínculos elásticos e amortecedores viscosos, sendo a equação de equilíbrio dinâmico resolvida através do método de Newmark. O esquema de acoplamento sólido-fluido adotado é o particionado convencional, que impõe condições de compatibilidade cinemáticas e de equilíbrio sobre a interface sólido-fluido, analisando ambos os meios de maneira sequencial. A discretização das equações governantes é realizada através do esquema explícito de dois passos de Taylor-Galerkin, aplicado no contexto da Análise Isogeométrica. Por fim, são analisados alguns problemas da Dinâmica de Fluidos Computacional, de onde se concluiu que os resultados obtidos são bastante consistentes com os fenômenos envolvidos, com as ferramentas exclusivas da Análise Isogeométrica, como o refinamento k, melhorando a convergência dos resultados. Para escoamentos bifásicos, verificou-se que o método Level Set obteve resultados bastante promissores apresentando, entretanto, uma dissipação numérica excessiva. Propõe-se, para estudos futuros, a elaboração de esquemas numéricos que conservem melhor o volume da fase líquida do escoamento. / The present work aims to development of a numerical formulation based on Isogeometric Analysis for the study of Fluid-Structure Interaction problems in applications involving rigid bodies submerged, considering incompressible Newtonian flows with free surface. The use of the Isogeometric Analysis allows unification between the preprocessing and analysis steps, improving then the continuity of the base functions employed, both in the spatial discretization and approximation of the variables in the system of equations. The fundamental flow equations are formed by the Navier-Stokes and the mass conservation, described by de pseudo-compressibility hypothesis, in an ALE (Arbitrary Lagrangean-Eulerian) kinematic formulation. The free surface consideration of the flow is handled treating the fluid like a two- phase medium, using the Level Set method. The rigid body considers nonlinearity in rotation, and restrictions represented by elastic springs and viscous dampers, with the dynamic equilibrium equation being resolved using the Newmark’s method. The solid-fluid coupling scheme is the conventional partitioned, which imposes kinematics and equilibrium compatibility conditions on the solid-fluid interface, analyzing both mediums in a sequential manner. The governing equations are discretized using the explicit two step Taylor-Galerkin method, applied in an Isogeometric Analisys context. Finally, some Computational Fluid Dinamics problems are analysed, from which it was concluded that the results obtained are quite consistent with phenomena involved, with the unique tools of Isogeometric Analysis, such as k-refinement, improving the convergence of the results. For biphasic flows, it was verified that the Level Set method obtained very promising results, presenting, however, an excessive numerical dissipation. For future studies, it is proposed the elaboration of numerical schemes that better preserve the volume of the liquid phase of the flow.

Le modèle de couche mince intégré sur l'épaisseur, Saint-Venant, utilisé classiquement pour simuler la propagation de laves torrentielles et coulées boueuses, repose sur plusieurs approximations concernant la forme des profils de vitesse en zones non-uniformes. Il est pourtant nécessaire d'utiliser ce type de modélisation, comme outil d'aide à la gestion des risques liés aux laves torrentielles. Nous proposons d'éprouver ses hypothèses, en observant une zone fortement non-uniforme, le front de coulées à surface libre et le champ de vitesse à l'intérieur de cette zone.En améliorant notre connaissance de l'évolution de la forme des profils de vitesse (de la dynamique interne) au front de coulées, nous cherchons à améliorer les modèles de couche mince. Cette thèse porte donc sur l'étude de la dynamique interne au front d'écoulements à surface libre de fluides newtoniens et viscoplastiques.Nous avons utilisé le dispositif du canal à fond mobile qui permet de générer des coulées stationnaires dans le référentiel de l'observateur au moyen d'un fond mobile remontant vers l'amont. Nous avons réalisé un travail technique sur ce canal et sur l'analyse des images pour pouvoir mesurer les champs de vitesse à haute résolution spatiale aux fronts de coulées à surface libre de fluides viscoplastiques. L'étude des fluides newtoniens a aussi été réalisée afin de valider les modèles et éprouver le dispositif expérimental.Nous avons comparé les résultats expérimentaux aux solutions théoriques de deux modèles de couche mince adaptés à la rhéologie de Herschel-Bulkley : le modèle classique de la lubrification, à la base du modèle de Saint-Venant et un modèle consistant à l'ordre 1 développé dans cette thèse. Le modèle consistant d'ordre 1 est la somme du modèle à l'ordre 0 (la lubrification) et de termes correctifs qui proviennent des contraintes normales et des termes d'inertie. Dans le cadre de notre configuration du fond mobile remontant vers l'amont, il est possible de déduire la forme du front en cherchant une solution de type onde progressive, sans passer par un modèle intégré dans l'épaisseur.Pour les fluides viscoplastiques, la structure classique du profil de vitesse, avec une zone cisaillée surmontée d'un plug non cisaillé est bien reconnaissable sur nos profils de vitesse en zone uniforme, et en zone faiblement variée. Mais à l'approche du front, cependant, la vitesse de surface augmente, les profils de vitesse expérimentaux deviennent cisaillés sur toute l'épaisseur, conduisant à la disparition du plug à proximité de la ligne de front.Le modèle de lubrification prédit l’existence d'un plug dans le front jusqu'à la ligne de contact, ce qui n'est pas observé expérimentalement. La vitesse de surface du modèle de lubrification augmente à l'approche du front, mais est largement sous-estimée par rapport à la vitesse de surface mesurée. Les vitesses de surface prédites par le modèle d'ordre 1 augmentent plus drastiquement au front, en meilleur accord avec les mesures que le modèle de lubrification. Pour certaines configurations expérimentales l'accord est même très bon. Remarquablement, le cisaillement des profils de vitesse à l'approche du front, observé expérimentalement, est aussi prédit par le modèle d'ordre 1.Les profils de vitesse présentent donc une évolution au front de coulées viscoplastiques en contradiction avec les hypothèses du modèle de Saint-Venant. Le modèle consistant d'ordre 1 permet d'améliorer les prédictions. Un modèle intégré dans l'épaisseur de type Saint-Venant basé sur les développements consistants d'ordre 1 est alors calculé, car il constitue l'étape nécessaire avant d'être intégré dans un outil de simulation opérationnel. / A depth-averaged model based on the thin-layer assumption, called Saint-Venant (Shallow-Water), is classically used to simulate the propagation and the spreading of debris and mud flows. It is based on several approximations concerning the shape of the velocity profile in non-uniform zones. We propose to test these hypotheses, examining a strongly non-uniform zone, the front of free-surface viscoplastic flows and the velocity field within this zone. By improving our knowledge about the internal dynamics in the front zone, we seek to improve the thin-layer models. This thesis therefore focuses on the study of the internal dynamics within the front of viscoplatic free-surface flows.We used the moving conveyor belt to generate stationary flows. We carried out a technical work on this set-up, and specific analysis of images obtained from the high-speed camera, in order to be able to measure velocity fields with a high resolution. The study of a Newtonian fluid was also carried out in order to validate the lubrication model and the experimental device.We compared experimental results to theoretical solutions of two thin-layer models taking into account the Herschel-Bulkley rheology: the classical model of lubrication, which is at the base of Saint-Venant model, and a consistent first-order model specifically developed in this thesis.The first-order model is equal to the zero-order model (lubrication), plus corrective terms derived from the normal stresses and inertia terms.In this study, for the purpose of comparison with our experimental results, we are interested in travelling-wave solutions. We are able to solve the shape of the front without using a depth-averaged model.Far from the front, experimental velocity profiles clearly display the characteristic 2-layer structure predicted by the lubrication solution, with constant values close to the free-surface (plug) and a sheared layer underneath. Closer to surge tip, the shape of experimental longitudinal velocity profilesthen begins to differ from the theoretical prediction. The 2-layer structure tends to disappear, and the profiles display shear across the whole depth ofthe flow. In this tip region, surface velocity also appears to increase faster than its theoretical counterpart. Surface velocity predicted by the first-order model increase more drastically in the tip region, in better agreement with the measurements than the lubrication model. The first-order model predicts a sheared velocity profile when approaching the front, as observed experimentally.The consistent first-order model then provides better predictions about internal dynamics than lubrication model. A depth-integrated model like Saint-Venant, based on consistent first-order developments is then calculated, as a first step before being integrated into an operational simulation tool.

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de problemas de interação fluido-estrutura (IFE) em aplicações envolvendo corpos rígidos submersos, onde escoamentos incompressíveis de fluidos Newtonianos com superfície livre são considerados. Propõe-se o emprego da Análise Isogeométrica por permitir a unificação entre os procedimentos de pré-processamento e análise, melhorando assim as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação da conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, em uma formulação cinemática ALE (Arbitrary Lagrangean- Eulerian). A consideração da superfície livre no escoamento se dá tratando o fluido como um meio bifásico, através do método Level Set. O corpo rígido apresenta não linearidade na rotação e restrições representadas por vínculos elásticos e amortecedores viscosos, sendo a equação de equilíbrio dinâmico resolvida através do método de Newmark. O esquema de acoplamento sólido-fluido adotado é o particionado convencional, que impõe condições de compatibilidade cinemáticas e de equilíbrio sobre a interface sólido-fluido, analisando ambos os meios de maneira sequencial. A discretização das equações governantes é realizada através do esquema explícito de dois passos de Taylor-Galerkin, aplicado no contexto da Análise Isogeométrica. Por fim, são analisados alguns problemas da Dinâmica de Fluidos Computacional, de onde se concluiu que os resultados obtidos são bastante consistentes com os fenômenos envolvidos, com as ferramentas exclusivas da Análise Isogeométrica, como o refinamento k, melhorando a convergência dos resultados. Para escoamentos bifásicos, verificou-se que o método Level Set obteve resultados bastante promissores apresentando, entretanto, uma dissipação numérica excessiva. Propõe-se, para estudos futuros, a elaboração de esquemas numéricos que conservem melhor o volume da fase líquida do escoamento. / The present work aims to development of a numerical formulation based on Isogeometric Analysis for the study of Fluid-Structure Interaction problems in applications involving rigid bodies submerged, considering incompressible Newtonian flows with free surface. The use of the Isogeometric Analysis allows unification between the preprocessing and analysis steps, improving then the continuity of the base functions employed, both in the spatial discretization and approximation of the variables in the system of equations. The fundamental flow equations are formed by the Navier-Stokes and the mass conservation, described by de pseudo-compressibility hypothesis, in an ALE (Arbitrary Lagrangean-Eulerian) kinematic formulation. The free surface consideration of the flow is handled treating the fluid like a two- phase medium, using the Level Set method. The rigid body considers nonlinearity in rotation, and restrictions represented by elastic springs and viscous dampers, with the dynamic equilibrium equation being resolved using the Newmark’s method. The solid-fluid coupling scheme is the conventional partitioned, which imposes kinematics and equilibrium compatibility conditions on the solid-fluid interface, analyzing both mediums in a sequential manner. The governing equations are discretized using the explicit two step Taylor-Galerkin method, applied in an Isogeometric Analisys context. Finally, some Computational Fluid Dinamics problems are analysed, from which it was concluded that the results obtained are quite consistent with phenomena involved, with the unique tools of Isogeometric Analysis, such as k-refinement, improving the convergence of the results. For biphasic flows, it was verified that the Level Set method obtained very promising results, presenting, however, an excessive numerical dissipation. For future studies, it is proposed the elaboration of numerical schemes that better preserve the volume of the liquid phase of the flow.

O presente trabalho tem por objetivo desenvolver uma formulação numérica baseada em Análise Isogeométrica para o estudo de problemas de interação fluido-estrutura (IFE) em aplicações envolvendo corpos rígidos submersos, onde escoamentos incompressíveis de fluidos Newtonianos com superfície livre são considerados. Propõe-se o emprego da Análise Isogeométrica por permitir a unificação entre os procedimentos de pré-processamento e análise, melhorando assim as condições de continuidade das funções de base empregadas tanto na discretização espacial do problema como na aproximação das variáveis do sistema de equações. O sistema de equações fundamentais do escoamento é formado pelas equações de Navier-Stokes e pela equação da conservação de massa, descrita segundo a hipótese de pseudo-compressibilidade, em uma formulação cinemática ALE (Arbitrary Lagrangean- Eulerian). A consideração da superfície livre no escoamento se dá tratando o fluido como um meio bifásico, através do método Level Set. O corpo rígido apresenta não linearidade na rotação e restrições representadas por vínculos elásticos e amortecedores viscosos, sendo a equação de equilíbrio dinâmico resolvida através do método de Newmark. O esquema de acoplamento sólido-fluido adotado é o particionado convencional, que impõe condições de compatibilidade cinemáticas e de equilíbrio sobre a interface sólido-fluido, analisando ambos os meios de maneira sequencial. A discretização das equações governantes é realizada através do esquema explícito de dois passos de Taylor-Galerkin, aplicado no contexto da Análise Isogeométrica. Por fim, são analisados alguns problemas da Dinâmica de Fluidos Computacional, de onde se concluiu que os resultados obtidos são bastante consistentes com os fenômenos envolvidos, com as ferramentas exclusivas da Análise Isogeométrica, como o refinamento k, melhorando a convergência dos resultados. Para escoamentos bifásicos, verificou-se que o método Level Set obteve resultados bastante promissores apresentando, entretanto, uma dissipação numérica excessiva. Propõe-se, para estudos futuros, a elaboração de esquemas numéricos que conservem melhor o volume da fase líquida do escoamento. / The present work aims to development of a numerical formulation based on Isogeometric Analysis for the study of Fluid-Structure Interaction problems in applications involving rigid bodies submerged, considering incompressible Newtonian flows with free surface. The use of the Isogeometric Analysis allows unification between the preprocessing and analysis steps, improving then the continuity of the base functions employed, both in the spatial discretization and approximation of the variables in the system of equations. The fundamental flow equations are formed by the Navier-Stokes and the mass conservation, described by de pseudo-compressibility hypothesis, in an ALE (Arbitrary Lagrangean-Eulerian) kinematic formulation. The free surface consideration of the flow is handled treating the fluid like a two- phase medium, using the Level Set method. The rigid body considers nonlinearity in rotation, and restrictions represented by elastic springs and viscous dampers, with the dynamic equilibrium equation being resolved using the Newmark’s method. The solid-fluid coupling scheme is the conventional partitioned, which imposes kinematics and equilibrium compatibility conditions on the solid-fluid interface, analyzing both mediums in a sequential manner. The governing equations are discretized using the explicit two step Taylor-Galerkin method, applied in an Isogeometric Analisys context. Finally, some Computational Fluid Dinamics problems are analysed, from which it was concluded that the results obtained are quite consistent with phenomena involved, with the unique tools of Isogeometric Analysis, such as k-refinement, improving the convergence of the results. For biphasic flows, it was verified that the Level Set method obtained very promising results, presenting, however, an excessive numerical dissipation. For future studies, it is proposed the elaboration of numerical schemes that better preserve the volume of the liquid phase of the flow.

No contexto do método MAC e baseado em esquemas de diferenças finitas, este trabalho apresenta três estudos: i) uma análise de estabilidade, ii) o desenvolvimento de técnicas implícitas e, iii) a construção de métodos de projeção para escoamentos com superfície livre. Na análise de estabilidade, o principal resultado mostra que o método de Crank-Nicolson torna-se condicionalmente estável quando aplicado para uma malha deslocada com a discretiza ção explícita das condições de contorno do tipo Dirichlet. Entretanto, o mesmo método com condições de contorno implícitas é incondicionalmente estável. Para obter métodos mais estáveis, formulações implícitas são desenvolvidas para a equação da pressão na superfície livre, derivada da condição de tensão normal. Esta estratégia resulta no acoplamento dos campos de velocidade e pressão, o que exige a introdução de novos métodos de projeção. Os métodos de projeção assim desenvolvidos resultam em novas metodologias para escoamentos com superfície livre que são apropriados para o tratamento de problemas com baixo número de Reynolds. Além disso, mostra-se que os métodos propostos podem ser aplicados para fluidos viscoelásticos. Novas estratégias são derivadas para obter métodos de projeção de segunda ordem de precisão para escoamentos com superfícies livres. Além dos resultados teóricos sobre a estabilidade de esquemas numéricos, técnicas implícitas e métodos de projeção, testes computacionais são realizados e comparados para consolidação da teoria apresentada. Os resultados numéricos são obtidos no sistema FREEFLOW. A eficiência e robustez das técnicas desenvolvidas neste trabalho são demonstradas na solução de problemas tridimensionais complexos com superfície livre e baixo número de Reynolds, incluindo os problemas do jato oscilante e do inchamento do extrudado / In the context of the MAC method and based on finite difference schemes, this work presents three studies: i) a stability analysis, ii) the development of implicit techniques, and iii) the construction of projection methods for free surface flows. In the stability analysis, the main result shows a precise stability restriction on the Crank-Nicolson method when one uses a staggered grid with Dirichlet explicit boundary conditions. However, the same method with implicit boundary conditions becomes unconditionally stable. In order to obtain more stable methods, implicit formulations are applied for the pressure equation at the free surface, which is derived from the normal stress condition. This approach results in a coupling of the velocity and pressure fields; hence new projection methods for free surface flows need to be developed. The developed projection methods result in new methodologies for low Reynolds number free surface flows. It is also shown that the proposed methods can be applied for viscoelastic fluids. New strategies are derived for obtaining second-order accurate projection methods for free surface flows. In addition to the theoretical results on the stability of numerical schemes, implicit techniques and projection methods, computational tests are carried out and the results compared to consolidate the theory. The numerical results are obtained by the FREEFLOW system. The eficiency and robustness of the techniques in this work are demonstrated by solving complex tridimensional problems involving free surface and low Reynolds numbers, including the jet buckling and the extrudate swell problems

Dans cette thèse, on s’intéresse à l’approximation numérique d’un modèle d’écoulement dispersif en eau peu profonde. Les applications visées par ce type de modélisation sont nombreuses (écoulement dans les océans, les rivières, etc) et cette thèse est motivée en particulier par les risques naturels et la production d’énergie renouvelable. Le modèle étudié a été dérivé par moyenne selon la verticale des équations d’Euler et prend en compte la pression non-hydrostatique. Il est alors nécessaire de résoudre un système de type incompressible; ce qui nous amène à résoudre une équation elliptique en pression. Nous proposons une méthode numérique pour résoudre le système dispersif avec topographie pour les modèles 1D et 2D. L’approche développée est basée sur un schéma de type prediction-correction, initialement introduit par Chorin-Temam pour les équations de Navier-Stokes. Nous définissons un cadre générique qui permet de concevoir un schéma valable en 1D et 2D et aussi de pouvoir augmenter l’ordre de précision. Ainsi, nous proposons une formulation variationnelle qui nous permet d’appliquer la méthode des éléments finis avec des choix d’espaces compatibles. Le travail effectué étant destiné à simuler des processus géophysiques réels, la méthode a été conçue pour pouvoir traiter les transitions de sol sec/mouillé et cette propriété a été confirmée par plusieurs tests numériques. Afin de valider la méthode, nous présentons la comparaison entre certaines solutions analytiques et leurs simulations numériques. / In this PhD thesis we are interested in the numerical approximation of a dispersive shallow water system, aimed at modeling the free surface flows (e.g. ocean and rivers) and motivated by applications for natural hazards and sustainable energy resources. This model is a depth-averaged Euler system and takes into account a non-hydrostatic pressure which brings crucial information for understanding the behavior of the flow, particularly when dispersion occur. We develop a numerical method for the one- and the two-dimensional dispersive shallow water system with a topography. The approach is based on a prediction-correction method initially introduced by Chorin-Temam, and we establish a global framework in order to easily increase the order of accuracy of the method. The prediction part leads to solving a shallow water system for which we use finite volume methods, while the correction part leads to solving a mixed problem in velocity and pressure. We propose a variational formulation of the mixed problem which allows us to apply a finite element method with compatible spaces. In this framework we establish compatible boundary conditions between the prediction part and the correction part. The method is performed for the one-dimensional model and for the two-dimensional problem on unstructured grids. In order to make the method practical for real geophysical cases, we have derived a scheme able to treat wet/dry interfaces and to this end we give many examples to test its performance. Moreover, we provide a comparison of simulated solutions with data from laboratory experiments.

<div>For centuries, fluid flows (hydrodynamics) and electromagnetic phenomena have interested scientists and laypeople alike. The earliest recording of the intersection of these two ideas, electro-hydrodynamics, was reported four centuries ago by William Gilbert who observed that static electricity generated from rubbed amber could ``attract" water. Today electrohydrodynamic phenomena are the underlying mechanisms driving the production of nano-fibers through electro-spinning, printing circuitry, and electrospraying, which John Fenn used in his Nobel prize winning work on electrospray ionization mass spectrometry. In all of these applications, a strong electric field is used to deform a liquid-gas interface (free surface) into a sharp conical tip. Unable to sustain these large interfacial stresses, a thin jet of fluid emerges from the tip of the cone and may subsequently break into a stream of smaller droplets. This tip-streaming phenomenon demands fundamental understanding of three canonical problems in fluid mechanics: electrified cones (Taylor cones), jets, and droplets. </div><div>In this thesis, the electrohydrodynamics of free surface flows are examined through both analytical and numerical treatment of the Cauchy momentum equations augmented with Maxwell's equations. Linear oscillations and stability of (inviscid) conducting charged droplets are examined in the presence of a solid ring shaped constraint. Here the constraint gives rise to an additional mode of oscillation---absent in the analysis of a free (unconstrained) droplet. Interestingly, the amount of charge necessary for instability, the Rayleigh charge limit, is unaltered by the constraint, but the mode of oscillation associated with instability changes. While all of the aforementioned applications involve electrified liquid-gas interfaces, recent experiments reveal a previously unknown type of streaming can occur for droplets suspended in another fluid. In these experiments, the suspending fluid is more conductive and an external electric field drives the intially spherical drop to adopt an oblate shape. Based on the viscosity ratio between the drop and suspending fluid, two different types of instability were observed: (i) if the drop is more viscous, then the drop forms a dimple at its poles and ruptures though its center, a phenomenon that is now referred to as dimpling, and (ii) if the suspending fluid is more viscous, then the drop adopts a lens-like shape and emits a sheet from its equator that subsequently breaks into a stream of rings and then tiny droplets, a phenomenon that is now called equatorial streaming. The physics of these two instabilities are far beyond the applicability of linear theory, requiring careful numerical analysis. Here steady-state governing equations are solved using the Galerkin finite element method (GFEM) to reveal the exact nature of these two instabilities and their dependence on the viscosity ratio. The fate of these drops once they succumb to instability is then analyzed by fully transient simulations.</div><div> Lastly, in a growing number of applications, the working fluid is non-Newtonian, and may even contain suspended solid particles. When non-Newtonian rheology is attributable to the presence of polymer, the dynamics is analyzed by means of a DEVSS-TG/SUPGFEM algorithm that is developed for simulating viscoelastic free surface </div><div>flows. When complex fluid rheology is due to the presence of suspended solid spherical particles, both early-time (linear) and asymptotic dynamics are uncovered by coupling the motion of the particles and Newtonian fluid implicitly in a GFEM fluid-structure interaction (FSI) algorithm. These novel algorithms are used to analyze the pinch-off dynamics of liquid jets and drops.</div>