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.

Dans cette thèse, nous étudions les propriétés des écoulements de fluides qui interagissent avec un corps rigide ou avec une source et un puits. Dans le cas d'un fluide visqueux incompressible qui satisfait les équations de Navier Stokes dans un domaine borné 2D, les solutions faibles de Leray-Hopf sont bien comprises. L'existence et l'unicité sont prouvées. De plus, les solutions sont continues en temps `a valeurs dans L 2 (Omega) et satisfont l’égalité d'énergie classique. Plus récemment, le problème d'un corps rigide en mouvement dans un fluide visqueux incompressible modélisé par les équations de Navier-Stokes couplées aux lois de Newton qui décrivent le mouvement du solide a également été abordé dans le cas où des conditions aux limites sans glissement ont été prescrites. Des résultats analogues concernant les solutions de Leray-Hopf ont également été démontrés dans ce contexte. Dans ce manuscrit, nous étudions le cas de conditions aux limites de Navier-Slip. Dans ce cadre, le résultat d'existence pour le système couplé a été prouvée par G'erard-Varet et Hillairet en 2014. Ici, nous montrons que les solutions sont continues en temps, qu'elles satisfont l’égalité d'énergie et qu’elles sont uniques. De plus, nous montrons un résultat d'existence des solutions faibles dans le cas d'un fluide incompressible visqueux auquel s'ajoute un corps rigide dans le cas où la vitesse du fluide a une partie orthonormale d'énergie infinie.Pour un fluide incompressible non visqueux modélisé par les équations d'Euler dans un domaine borné 2D, le cas où le fluide est autorisé à entrer et à sortir de la frontière a été traité par Judovic qui a introduit certaines conditions limites consistant à prescrire la composante normale de la vitesse et de la vorticité entrante. Dans ce manuscrit, nous considérons un domaine borné qui possède deux trous. L'un d'eux est une source, ce qui signifie que le fluide est autorisé à entrer dans le domaine et l'autre est un puits où le fluide peut sortir. En particulier, nous établissons les équations limites vérifiées par le fluide lorsque la source et le puits se contractent en deux points différents. Le système limite est caractérisé par un point source/puits et un point vortex en chacun des deux points où les trous se sont contractés. / In this thesis, we investigate properties of incompressible flows that interact with a rigid body or a source and a sink. In the case of an incompressible viscous fluid that satisfies the Navier Stokes equations in a 2D bounded domain well-posedness of Leray-Hopf weak solutions is well-understood. Existence and uniqueness are proved. Moreover solutions are continuous in time with values in L 2 (Omega) and they satisfy the energy equality. Recently the problem of a rigid body moving in a viscous incompressible fluid modeled by the Navier-Stokes equations coupled with the Newton laws that prescribe the motion of the solid, was also tackled in the case where the no-slip boundary conditions were imposed. And the correspondent well-posedness result for Leray-Hopf type weak solutions was proved. In this manuscript we consider the case of the Navier-slip boundary conditions. In this setting, the existence result for the coupled system was proved by G'erard-Varet and Hillairet in 2014. Here, we prove that solutions are continuous in time, that they satisfy the energy equality and that they are unique. Moreover we show an existence result for weak solutions of a viscous incompressible fluid plus rigid body system in the case where the fluid velocity has an orthoradial part of infinite energy.For an inviscid incompressible fluid modelled by the Euler equations in a 2D bounded domain, the case where the fluid is allowed to enter and to exit from the boundary was tackled by Judovic who introduced some conditions which consist in prescribing the normal component of the velocity and the entering vorticity. In this manuscript we consider a bounded domain with two holes, one of them is a source which means that the fluid is allowed to enter in the domain and the other is a sink from where the fluid can exit. In particular we find the limiting equations satisfied by the fluid when the source and the sink shrink to two different points. The limiting system is characterized by a point source/sink and a point vortex in each of the two points where the holes shrunk.

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.

In this thesis study, a least-squares finite element method (LSFEM) based conjugate heat transfer solver was developed. In the mentioned solver, fluid flow and heat transfer computations were performed separately. This means that the calculated velocity values in the flow calculation part were exported to the heat transfer part to be used in the convective part of the energy equation. Incompressible Navier-Stokes equations were used in the flow simulations. In conjugate heat transfer computations, it is required to calculate the heat transfer in both flow field and solid region. In this study, conjugate behavior was accomplished in a fully coupled manner, i.e., energy equation for fluid and solid regions was solved simultaneously and no boundary conditions were defined on the fluid-solid interface. To assure that the developed solver works properly, lid driven cavity flow, backward facing step flow and thermally driven cavity flow problems were simulated in three dimensions and the findings compared well with the available data from the literature. Couette flow and thermally driven cavity flow with conjugate heat transfer in two dimensions were modeled to further validate the solver. Finally, a microchannel conjugate heat transfer problem was simulated. In the flow solution part of the microchannel problem, conservation of mass was not achieved. This problem was expected since the LSFEM has problems related to mass conservation especially in high aspect ratio channels. In order to overcome the mentioned problem, weight of continuity equation was increased by multiplying it with a constant. Weighting worked for the microchannel problem and the mass conservation issue was resolved. Obtained results for microchannel heat transfer problem were in good agreement in general with the previous experimental and numerical works. In the first computations with the solver / quadrilateral and triangular elements for two dimensional problems, hexagonal and tetrahedron elements for three dimensional problems were tried. However, since only the quadrilateral and hexagonal elements gave satisfactory results, they were used in all the above mentioned simulations.

This thesis is concerned with the development of numerical techniques to simulate compressible multi-phase flows, in particular a high-accuracy numerical approach with mesh adaptivity. The Discontinuous Galerkin (DG) method was chosen as the framework for this work for being characterized for its high-order of accuracy -thus low numerical diffusion- and being compatible with mesh adaptivity due to its locality. A DG solver named DiGGIT (Discontinuous Galerkin at the Georgia Institute of Technology) has been developed and several aspects of the method have been studied. The Local Discontinuous Galerkin (LDG) method -an extension of DG for equations with high-order derivatives- was extended to solve multiphase flows using Diffused Interface Methods (DIM). This multi-phase model includes the convection of the volume fraction, which is treated as a Hamilton-Jacobi equation. This is the first study, to the author's knowledge, in which the volume fraction of a DIM is solved using the DG and the LDG methods. The formulation is independent of the Equation of State (EOS) and it can differ for each phase. This allows for a more accurate representation of the different fluids by using cubic EOSs, like the Peng-Robinson and the van der Waals models. Surface tension is modeled with a new numerical technique appropriate for LDG. Spurious oscillations due to surface tension are common to all the capturing schemes, and this new approach presents oscillations comparable in magnitude to the most common schemes. The moment limiter (ML) was generalized for non-uniform grids with hanging nodes that result from adaptive mesh refinement (AMR). The effect of characteristic, primitive, or conservative decomposition in the limiting stage was studied. The characteristic option cannot be used with the ML in multi-dimensions. In general, primitive variable decomposition is a better option than with conservative variables, particularly for multiphase flows, since the former type of decomposition reduces the numerical oscillations at material discontinuities. An additional limiting technique was introduced for DIM to preserve positivity while minimizing the numerical diffusion, which is especially important at the interface. The accuracy-preserving total variation diminishing (AP-TVD) marker for ``troubled-cell' detection, which uses an averaged-derivative basis, was modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique. Furthermore, a new error estimator was proposed to determine where to refine or coarsen the grid. This estimator was compared against other estimator used in the literature and it showed an improved performance. In order to provide equal order of accuracy in time as in space, the commonly used 3rd-order TVD Runge-Kutta (RK) scheme in the DG method was replaced in some cases by the Spectral Deferred Correction (SDC) technique. High orders in time were shown to only be required when the error in time is significant. For instance, convection-dominated compressible flows require for stability a time step much smaller than is required for accuracy, so in such cases 3rd-order TVD RK resulted to be more efficient than SDC with higher orders. All these new capabilities were included in DiGGIT and have provided a generalized approach capable of solving sub- and super-critical flows at sub- and super-sonic speeds, using a high-order scheme in space and time, and with AMR. Canonical test cases are presented to verify and validate the formulation in one, two, and three dimensions. Finally, the solver is applied to practical applications. Shock-bubble interaction is studied and the effect of the different thermodynamic closures is assessed. Interaction between single-drops and a wall is simulated. Sticking and the onset of splashing are observed. In addition, the solver is used to simulate turbulent flows, where the high-order of accuracy clearly shows its benefits. Finally, the methodology is challenged with the simulation of a liquid jet in cross flow.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

An axisymmetric, Navier-Stokes finite volume flow solver, which uses Harten, Lax and van Leer (HLL) and Harten, Lax and van Leer&ndash / Contact (HLLC) upwind flux differencing scheme for spatial and uses Runge-Kutta explicit multi-stage time stepping scheme for temporal discretization on unstructured meshe is developed. Developed solver can solve the compressible axisymmetric flow. The spatial accuracy of the solver can be first or second order accurate. Second order accuracy is achieved by piecewise linear reconstruction. Gradients of flow variables required for piecewise linear reconstruction are calculated by Green-Gauss theorem. Baldwin-Lomax turbulent model is used to compute the turbulent viscosity. Approximate Riemann solver of HLL and HLLC implemented in solver are validated by solving a cylindrical explosion case. Also the solver&rsquo / s capability of solving unstructured, multi-zone domain is investigated by this problem. First and second order results of solver are compared by solving the flow over a circular bump. Axisymmetric flow in solid propellant rocket motor is solved in order to validate the axisymmetric feature of solver. Laminar flow over flat plate is solved for viscous terms validation. Turbulent model is studied in the flow over flat plate and flow with mass injection test cases.

A fully automated Cartesian/Quad grid generator and laminar flow solver have been developed for external flows by using C++. After defining the input geometry by nodal points, adaptively refined Cartesian grids are generated automatically. Quadtree data structure is used in order to connect the Cartesian cells to each other. In order to simulate viscous flows, body-fitted quad cells can be generated optionally. Connectivity is provided by cut and split cells such that the intersection points of Cartesian cells are used as the corners of quads at the outmost row. Geometry based adaptation methods for cut, split cells and highly curved regions are applied to the uniform mesh generated around the geometry. After obtaining a sufficient resolution in the domain, the solution is achieved with cellcentered approach by using multistage time stepping scheme. Solution based grid adaptations are carried out during the execution of the program in order to refine the regions with high gradients and obtain sufficient resolution in these regions. Moreover, multigrid technique is implemented to accelerate the convergence time significantly. Some tests are performed in order to verify and validate the accuracy and efficiency of the code for inviscid and laminar flows.