Les écoulements turbulents à phases séparées sont présents dans de très nombreuses applications. Cependant, la simulation de tels écoulements avec une interface déformable constitue l'un des problèmes les plus complexes de la mécanique des fluides numérique. La prise en compte du bilan des contraintes normales est au cœur du problème de déformation de l'interface. Dans le travail présenté ici, nous développons un algorithme permettant de simuler des écoulements diphasiques incompressibles et turbulents en suivant le déplacement de l'interface par une approche lagrangienne. Les équations de Navier-Stokes instationnaires écrites en variables vitesse-pression sont résolues dans les deux phases en utilisant des maillages curvilignes orthogonaux. Dans un premier temps, nous introduisons un schéma de raccordement des vitesses tangentielles et des cisaillements. Ce schéma est appliqué afin de simuler l'interaction de deux écoulements turbulents séparés par une interface plane. La turbulence est traitée par une approche de simulation des grandes échelles utilisant un modèle dynamique. Un algorithme original est ensuite développé dans le but de satisfaire de façon non-itérative à la fois la continuité des vitesses normales et des contraintes normales sur l'interface et l'incompressibilité dans les deux phases. Différentes simulations d'écoulements diphasiques avec interface déformable sont réalisées afin de valider ces développements. / Turbulent incompressible two-phase separated flows are present in many applications. However, simulation of such flows with a moving interface is one of the most challenging problems in todays computational fluid dynamics. Taking properly into account the normal stress budget accross the interface is the main difficulty of moving interface problems. This work deals with the development of a boundary-fitted method for computing turbulent incompressible two-phase flows. The interface displacement is achieved through a Lagrangian approach. The unsteady Navier-Stokes equations written in a velocity- ressure formulation are solved within the two phases using an orthogonal curvilinear grid. In a first step, we introduce a scheme allowing tangential velocities and shear stresses to match across the interface. We apply this technique to compute the countercurrent flow generated by two streams separated by a plane interface. This scheme is then applied to compute various situations involving the interaction between two turbulent flows separated by a flat interface. The turbulence is treated by using the Large Eddy Simulation approach with a dynamic model. An original algorithm is then developed to satisfy without any internal iteration the continuity of normal velocities and stresses across the interface and the incompressibility condition within both phases. Several simulations of two-phase flows with a moving interface are carried out to validate these developments.

Hot wire anemometers have been used in several wind velocity sensors deployed in Mars. They are based in keeping the temperature of a surface at a constant value, above the ambient. This is done by means of a heater controlled with an electronic system. The cooling rate of each point at the sensor surface can be used to calculate the wind velocity and direction. However, due to turbulent fluctuations, the cooling rate is not constant even in the case of constant velocity. Moreover, RANS simulations cannot estimate such fluctuations as they only provide an estimation of the averaged flow field. The goal of this work has been to estimate such fluctuations and the e↵ect they might have on the sensor readings. To do so, the turbulent cooling rate (Nusselt number) of a sensor with a generic shape, under the typical conditions to be find in Mars, has been simulated using high performance LES (Large Eddy Simulation) simulations and compared with RANS and URANS simulations.

Cette thèse est consacrée à l’analyse mathématique du problème du mouvement d’un nombre fini de corps rigides homogènes au sein d’un fluide visqueux incompressible homogène. Les fluides visqueux sont classés en deux catégories: les fluides newtoniens et les fluides non newtoniens. En premier lieu, nous considérons le système formé par les équations de Navier Stokes incompressible couplées aux lois de Newton pour décrire le mouvement de plusieurs disques rigides dans un fluide newtonien visqueux homogène dans l’ensemble de l’espace R^2. Nous montrons que ce problème est bien posé jusqu’à l’apparition de la première collision. Ensuite, nous éliminons tous les types de contacts pouvant survenir si le domaine fluide reste connexe à tout moment. Avec cette hypothèse, le système considéré est globalement bien posé. Dans la deuxième partie de cette thèse, nous montrons la non-unicité des solutions faibles au problème d’interaction fluide-solide 3D, dans le cas d’un fluide newtonien, après collision. Nous montrons qu’il existe des conditions initiales telles que nous pouvons étendre les solutions faibles après le temps pour lequel le contact a eu lieu de deux manières différentes. Enfin, dans la dernière partie, nous étudions le mouvement bidimensionnel d’un nombre fini de disques immergés dans une cavité remplie d’un fluide viscoélastique tel que des solutions polymériques. Les équations de Navier Stokes incompressible sont utilisées pour modéliser le solvant, dans lesquelles un tenseur de contrainte élastique supplémentaire apparaît comme un terme source. Dans cette partie, nous supposons que le tenseur de contrainte supplémentaire satisfait la loi différentielle d’Oldroyd ou sa version régularisée. Dans les deux cas, nous prouvons l’existence et l’unicité des solutions fortes locales en temps du problème considéré. / This thesis is devoted to the mathematical analysis of the problem of motion of afinite number of homogeneous rigid bodies within a homogeneous incompressible viscous fluid. Viscous fluids are classified into two categories: Newtonian fluids, and non-Newtonian fluids. First, we consider the system formed by the incompressible Navier-Stokes equations coupled with Newton’s laws to describe the movement of several rigid disks within a homogeneous viscous Newtonian fluid in the whole space R^2. We show the well-posedness of this system up to the occurrence of the first collision. Then we eliminate all type of contacts that may occur if the fluid domain remains connected at any time. With this assumption, the considered system is well-posed globally in time. In the second part of this thesis, we prove the non-uniqueness of weak solutions to the fluid-rigid body interaction problem in 3D in Newtonian fluid after collision. We show that there exist some initial conditions such that we can extend weak solutions after the time for which contact has taken place by two different ways. Finally, in the last part, we study the two-dimensional motion of a finite number of disks immersed in a cavity filled with a viscoelastic fluid such as polymeric solutions. The incompressible Navier–Stokes equations are used to model the flow of the solvent, in which the elastic extra stress tensor appears as a source term. In this part, we suppose that the extra stress tensor satisfies either the Oldroyd or the regularized Oldroyd constitutive differential law. In both cases, we prove the existence and uniqueness of local-in-time strongsolutions of the considered moving-boundary problem.

This study involves the modeling of small cold-gas (N2) thrusters nozzle and plume flows, their interactions with spacecraft surfaces and the induced pressure environment. These small cold-gas thrusters were used for pitch, yaw and roll control and were mounted on the bottom of the conical Environmental Monitor Payload (EMP) suborbital spacecraft. The pitch and yaw thrusters had 0.906 mm throat diameter and 4.826 mm exit diameter, while the roll thrusters had 1.6 mm throat diameter and 5.882 mm exit diameter. During thruster firing, at altitudes between 670 km and 1200 km, pressure measurements exhibited non-periodic pulses (Gatsonis et al., 1999). The pressure sensor was located inside the EMP and was connected to it's sidewall with a 0.1-m long, 0.022-m diameter tube and the pressure pulses appeared instantaneously with the firings for thrusters without a direct line-of-sight with the sensor entrance. Preliminary analysis showed that the plume of these small EMP thrusters undergoes transition from continuous to rarefied. Therefore, nozzle and plume simulations are performed using a combination of Navier-Stokes and Direct Simulation Monte Carlo codes. This study presents first a validation of the Navier-Stokes code Rampant used for the continuous EMP nozzle and plume simulations. The first Rampant validation example involves a two-dimensional axisymetric freejet expansion and is used to demonstrate the use of Bird's breakdown parameter. Results are compared favorably with those of Bird (1980) obtained through the method of characteristics. The second validation example involves three-dimensional plume simulations of a NASA thruster. This nitrogen nozzle has a throat diameter of 3.18 mm, an exit diameter of 31.8 mm, half-angle of 20 degrees, stagnation temperature of 699 K, stagnation pressure of 6,400 Pa. Simulation results are compared favorably with previous Navier-Stokes and Direct Simulation Monte Carlo numerical work. The third validation example involves three-dimensional simulations of Rothe's (1970) nozzle that has a throat diameter of 2.5 mm, an exit diameter of 20.3 mm, half-angle of 20 degrees, operating at stagnation temperature of 300 K and pressure of 1975 Pa. Numerical results also compared favorably to experimental data. The combined Navier-Stokes/DSMC approach and the EMP simulation results are presented and discussed. The continuous part of the EMP nozzle and plume flow is modeled using the three-dimensional Navier-Stokes Rampant code. The Navier-Stokes domain includes the geometry of the nozzle and the EMP base until transition of the continuous flow established by Bird's breakdown parameter. The rarefied part of the plume flow is modeled using the Direct Simulation Monte Carlo code DAC. Flowfield data obtained inside the breakdown surface from the Navier-Stokes simulation are used as inputs to the DSMC simulations. The DSMC domain includes the input surface and the EMP spacecraft geometry. The combined Navier-Stokes/DSMC simulations show the complex structure of the plume flow as it expands over the EMP surfaces. Plume reflection and backflow are demonstrated. The study also summarizes findings presented by Gatsonis et al. (2000), where the DSMC predictions at the entrance of the pressure sensor are used as inputs to a semi-analytical model to predict the pressure inside the sensor. It is shown that the pressure predictions for the pitch/yaw thrusters are close to the measurements. The plume of a pitch or yaw thruster reaches the pressure sensor after expanding on the EMP base. The pressure predicted for the roll thruster is larger that the measured. This is attributed to the uncertainty in the roll thruster location on the EMP base resulting, in the simulation, in a component of direct flow to the sensor.

O trabalho desenvolvido nesta tese propõe a aplicação do método da expansão em funções hierárquicas elaborado por Zienkiewics e Morgan (1983), para a solução das equações de conservação da massa (continuidade), conservação da quantidade de movimento (Navier-Stokes) e conservação da energia, para fluidos compressíveis em duas dimensões e em alta velocidade. Esse método consiste no emprego do método de elementos finitos utilizando a formulação Petrov-Galerkin, conhecido como SUPG (“Streamline Upwind Petrov-Galerkin"), desenvolvido por Brooks e Hughes (1982), aplicado em conjunto com uma expansão das variáveis em funções hierárquicas. A fim de testar e validar o método numérico proposto, assim como o programa computacional elaborado, foram simulados alguns casos conhecidos da literatura. Os casos estudados foram os seguintes: teste de Continuidade; teste de convergência e estabilidade; problema do degrau de temperatura e problema do choque oblíquo, onde o objetivo desse último caso era, basicamente, verificar a captura da onda de choque pelo método numérico desenvolvido. Através dos casos estudados e em função dos resultados obtidos nas simulações realizadas, conclui-se que o objetivo desse trabalho foi alcançado de maneira satisfatória, pois os resultados obtidos com o método desenvolvido nesse trabalho foram qualitativamente e quantitativamente bons, quando comparados com os resultados teóricos. / The Thesis develops a new application for the Hierarchical Function Expansion Method, proposed by Zienkiewics and Morgan (1983), for the solution of the Navier-Stokes equations for compressible fluids in two dimensions and in high velocity. This method is based on the finite elements method using the Petrov-Galerkin formulation, know as, SUPG (Streamline Upwind Petrov-Galerkin) developed by Brooks and Hughes (1982), and applied in conjunction with the expansion of the variables into hierarchical functions. To test and validate the numerical method proposed as well as the computational program developed some cases whose theoretical solution are known simulated. These cases are the following: continuity test; stability and convergence test; temperature step problem; and several oblique shocks. The objective of the last cases is basically to verify the capture of the shock wave by the method developed. The results obtained in the simulations of the cases performed with the proposed method were good both qualitatively and quantitatively when compared with the teorethical solutions. This allows us to conclude that the objective of this Thesis was satisfactorily reached.

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L_∞(-1; 0; L^{3, β}(B(1) ⋂ ℝ³ ₊)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ³ ₊×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T||v(·, t)||_{L^3,β(ℝ³ ₊)} = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and Šverák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ³, with solenoidal initial data in the critical Besov space ?^-1/4_4,∞(ℝ³), which has certain continuity properties with respect to weak* convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in ℝ³×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T ||v(·, t)||_{L₃(ℝ³)} = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

No description available.

O trabalho desenvolvido nesta tese propõe a aplicação do método da expansão em funções hierárquicas elaborado por Zienkiewics e Morgan (1983), para a solução das equações de conservação da massa (continuidade), conservação da quantidade de movimento (Navier-Stokes) e conservação da energia, para fluidos compressíveis em duas dimensões e em alta velocidade. Esse método consiste no emprego do método de elementos finitos utilizando a formulação Petrov-Galerkin, conhecido como SUPG (“Streamline Upwind Petrov-Galerkin”), desenvolvido por Brooks e Hughes (1982), aplicado em conjunto com uma expansão das variáveis em funções hierárquicas. A fim de testar e validar o método numérico proposto, assim como o programa computacional elaborado, foram simulados alguns casos conhecidos da literatura. Os casos estudados foram os seguintes: teste de Continuidade; teste de convergência e estabilidade; problema do degrau de temperatura e problema do choque oblíquo, onde o objetivo desse último caso era, basicamente, verificar a captura da onda de choque pelo método numérico desenvolvido. Através dos casos estudados e em função dos resultados obtidos nas simulações realizadas, conclui-se que o objetivo desse trabalho foi alcançado de maneira satisfatória, pois os resultados obtidos com o método desenvolvido nesse trabalho foram qualitativamente e quantitativamente bons, quando comparados com os resultados teóricos. / The Thesis develops a new application for the Hierarchical Function Expansion Method, proposed by Zienkiewics and Morgan (1983), for the solution of the Navier-Stokes equations for compressible fluids in two dimensions and in high velocity. This method is based on the finite elements method using the Petrov-Galerkin formulation, know as, SUPG (Streamline Upwind Petrov-Galerkin) developed by Brooks and Hughes (1982), and applied in conjunction with the expansion of the variables into hierarchical functions. To test and validate the numerical method proposed as well as the computational program developed some cases whose theoretical solution are known simulated. These cases are the following: continuity test; stability and convergence test; temperature step problem; and several oblique shocks. The objective of the last cases is basically to verify the capture of the shock wave by the method developed. The results obtained in the simulations of the cases performed with the proposed method were good both qualitatively and quantitatively when compared with the teorethical solutions. This allows us to conclude that the objective of this Thesis was satisfactorily reached.

A particulate model is developed to analyze the effects of steady state and transient seepage of water through a randomly-packed coarse-grained soil as an improvement to conventional seepage analysis based on continuum models. In the new model the soil skeleton and pore water are volumetrically coupled. In the first phase of the study, the concept of relative density has been used to define different compaction levels of the soil layers of a completely saturated pavement filter system and observe the seepage response to compaction. First, Monte-Carlo simulation is used to randomly pack discrete spherical particles from a specified Particle Size Distribution (PSD) to achieve a desired relative density based on the theoretical minimum and maximum void ratios. Then, a water pressure gradient is applied across one two-layer filter unit to trigger water seepage. The pore water motion is idealized using Navier Stokes (NS) equations which also incorporate drag forces acting between the water and soil particles. The NS equations are discretized using finite differences and applied to discrete elements in a staggered, structured grid. The model predicted hydraulic conductivities are validated using widely used equations. The critical water velocities, hydraulic gradients and flow within the xi saturated soil layers are identified under both steady state and transient conditions. Significantly critical transient conditions seem to develop. In the second phase of the study the model is extended to analyze the confined flow through a partly saturated pavement layer and unconfined flow from a retention pond into the surrounding saturated granular soil medium. In partly saturated soil, the water porosity changes resulting from water flow is updated using the Soil Water Characteristics Curve (SWCC) of the soil. The results show how complete saturation develops due to water flow following the water porosity Vs pressure trend defined by the SWCC. Finally, the model is used to predict the gradual reduction in the water level of a retention pond and the location of the free-surface. The free-surface is determined by differentiating the wet and dry zones based on the Heaviside step function modified NS equations.

Ce travail de thèse porte sur l'étude d'un système d'interaction fluide-structure. Nous en traitons de nombreux aspects allant de sa modélisation jusqu'à l'étude de sa stabilisation et de sa simulation numérique. Le premier chapitre du manuscrit aborde la modélisation du système ainsi que l'existence de solutions fortes en temps petits. Le fluide est représenté par les équations de Navier-Stokes incompressibles. La structure est déformable et dépend d'un nombre fini de paramètres. Nous obtenons ses équations en appliquant un principe des travaux virtuels. Le système d'équations final est non linéaire. Nous prouvons l'existence locale d'une solution à ce système, dans un premier temps sur le système linéarisé autour de l'état nul. Puis, nous prouvons l'existence de solutions en temps petits au système non linéaire grâce à un argument de point fixe. Le deuxième chapitre traite de la stabilisation par feedback autour d'un état stationnaire non nul du système présenté dans le Chapitre 1. L'opérateur de feedback est déterminé à partir de l'analyse du problème linéarisé autour de l'état stationnaire et de la résolution d'une équation de Riccati. Le résultat de stabilisation portant sur le système non linéaire requiert des données petites et est obtenu par un argument de point fixe. Le troisième chapitre se concentre sur les aspects numériques de ce problème. La construction de l'opérateur de feedback correspond à la version discrétisée de celle proposée dans le Chapitre 2. Le système fluide-structure est simulé en utilisant une méthode de domaines fictifs. / This PhD thesis deals with the study of a fluid-structure interaction system. We are interested in several aspects such as modelling, stabilization and numerical simulation. In the first chapter of the manuscript, we show the modelling of the system and prove the existence of strong solutions in small times. The fluid is modelled by the incompressible Navier- Stokes equations. The structure is deformable and depends on a finite number of parameters. The equations are obtained with a virtual work principle. The final system of equations is nonlinear. We prove local existence of a solution to this system, first on the linearized system. Then, existence of solutions in small times to the full nonlinear system is obtained with a fixed point argument. In the second chapter, we prove feedback stabilization of the problem around a non-null stationary state. The feedback operator is computed with the solution to a Riccati equation obtained by the analysis of the linearized problem around the stationary state. The stabilization result holds on the full nonlinear system and requires small data. It is proven by a fixed point argument. In the third chapter, we focus on the numerical aspects of the problem. The feedback operator used corresponds to a discretization of the feedback operator of Chapter 2. The solution to the full nonlinear system is computed by the use of a fictitious domain method.