Magnetohydrodynamics describes the motions of an electrically conducting fluid under the influence of magnetic fields. Such flows are encountered in a large variety of applications, from steel industry to heat exchangers of nuclear fusion reactors. <p><p>Here we are concerned with situations where the magnetic field is relatively strong and the flow manifests turbulent motions. The interaction of the fluid with the electromagnetic field is still insufficiently understood and efficient predicting methods are lacking. Our goal is to provide more insight on this problem by making heavy use of numerical methods. In this work, two different classes of problem are investigated. <p><p>First we consider that the turbulent character of the fluid is well developed and that solid boundaries are sufficiently far away to be completely neglected. The main effects of a strong magnetic field in that case are to damp the motion and to homogenize the flow along its direction, leading to a quasi two dimensional state. Using numerical simulations we have studied the dynamics of the flow in Fourier space and in particular the non linear energy transfers between turbulent eddies. Further we investigated the scale-by-scale anisotropy and compared various methods to address this quantity. Finally, the evolution of a passive scalar embedded in the flow was analyzed and it turned out that the characteristic anisotropy of the velocity field is reflected in the distribution of the scalar quantity. <p><p>In the second problem, the flow in a duct of square cross section subject to a transverse magnetic field has been considered. Here, unlike in the previous situation, the magnetic field has globally a destabilizing effect on the flow, because of the strong inhomogeneities it produces. For instance, high velocity regions develop along the walls that are parallel to the magnetic field. There, we are mostly interested in the possible development of persistent time-dependent fluctuations. It is observed that the transition between laminar and turbulent regimes occurs through at least two distinct bifurcations. The first one takes place at moderate Reynolds number and is characterized by highly organized fluctuations. The second is encountered at higher Reynolds number and presents very strong and localized disturbances.<p>/Il existe un grand nombre d'applications industrielles dans lesquelles un écoulement de métal liquide est soumis à un champ magnétique. La production d'acier par coulée continue, la fabrication de matériaux semi-conducteurs ou encore les échan-geurs de chaleur des futurs réacteurs à fusion nucléaire en sont de bons exemples. L'interaction du liquide conducteur avec le champ magnétique est à l'origine de nombreux phénomènes inhabituels en hydrodynamique classique et doit dès lors être décrite par la magnétohydrodynamique (ou MHD en abrégé). Le but de ce travail est d'étudier la physique de ces interactions, en se basant sur la résolution numérique des équations qui les gouvernent.<p><p>Plusieurs aspects du problème ont été considérés indépendamment. Tout d'abord, l'étude de la turbulence homogène a permis de mettre en evidence les comportements du fluide loin de toute paroi solide. Ceci est mis un oeuvre dans un domaine spatial périodique, où les variables sont représentées par leur série de Fourier. L'influence du champ magnétique dans ce cas consiste à dissiper les fluctuations turbulentes et à rendre le champ de vitesse anisotrope. Les principaux résultats obtenus dans ce cadre concernent la distribution ainsi que le transfert d'énergie dans l'espace spectral, l'anisotropie des différentes échelles turbulentes de l'écoulement ainsi que le transport d'un scalaire passif au sein du fluide. <p><p>Dans un deuxième temps, le travail a porté sur l'écoulement dans une conduite rectangulaire soumise à un champ magnétique et dont les parois sont conductrices d'électricité. La particularité de cet écoulement réside dans les zones de vitesse élevées qui se développent le long des parois parallèles au champ magnétique. Celles-ci donnent lieu à un intense cisaillement qui a généralement pour effet de rendre l'écoulement instable. La simulation numérique de ce problème a permis l'étude des instabilités au sein du fluide et de la transition du régime laminaire vers la turbulence. <p> / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

In our everyday life, turbulence is an omnipresent phenomenon and yet remains poorly understood. Its random and chaotic nature makes it a subject almost impossible to treat from the mathematical point of view and, at present, there<p>is no real prospect of a simple analytic theory. Scientists have therefore regarded the numerical simulation as an alternative to compute the relevant properties of turbulent flows. In this context, our thesis aims at developing and using accurate computational methods, namely pseudo-spectral methods, for studying hydrodynamic (1st part) and magnetohydrodynamic (2nd part) turbulence.<p><p>In the hydrodynamic part, Chapter I introduces the governing equations of fluid mechanics as well as the main issues related to the numerical study of turbulent flows. In particular, the Direct Numerical Simulations (DNS) of turbulence, in which accurate numerical solutions of the Navier-Stokes equations are obtained, are shown to be limited to moderately turbulent flows.<p>Chapter II introduces the Large Eddy Simulation (LES) technique which aims at simulating highly turbulent flows and which is based on a separation of scales.<p>In practice, it consists of simulating the large - resolved - scales of the flow explicitly while modelling the small - unresolved - scales. Two different approaches for modelling the kinetic energy of the unresolved scales are proposed and their respective advantages and drawbacks are discussed.<p>Chapter III is devoted the study of the mixing-layer using both DNS and LES. It consists of an inhomogeneous turbulent flow which has been studied experimentally and for which well-documented measurements are available. A highly accurate DNS mimicking the same experiment has been produced. It allows to study the inhomogeneity and anisotropy properties of this flow. Also, LES of the same flow, using different models, have been evaluated. In Chapter IV, we explore a pseudo-spectral method to investigate turbulence in a pipe. In this case, the method has to take into account two additional difficulties: i) the presence of the boundary and ii) the axis singularity. We detail how to circumvent these issues.<p><p>The second part of the thesis is devoted to magnetohydrodynamic (MHD) turbulence. It concerns phenomena where electrically conducting flows interact with electromagnetism and for which governing equations are derived in Chapter V. In Chapter VI, a detailed analysis of the energy transfers between the magnetic and velocity fields is performed thanks to a high resolution database of homogeneous MHD turbulence. It provides some insights to understand the physics of the nonlinear interactions and is also a valuable diagnostic in the framework of LES modelling. Finally, the inhomogeneous configuration studied in Chapter III has been extended to MHD. Several statistics related to the kinetic and magnetic energies are measured and LES of this flow are performed and presented in Chapter VII. / Doctorat en sciences, Spécialisation physique / info:eu-repo/semantics/nonPublished

Le couplage important entre les différentes échelles d’un écoulement est une des caractéristiques prin-cipales des turbulences. Cela est exprimé mathématiquement par les termes non linéaires présents dans les équations d’équilibre de l’écoulement, dominants en dynamique turbulente. En magnétohy-drodynamique (MHD), la force de Lorentz influe sur l’équation de conservation de l’impulsion et le nombre de termes non linéaires passe à quatre au lieu d’un seul pour un fluide non conducteur.<p>L’objectif principal de cette thèse est d’analyser le transport d’énergie inter-échelles en utilisant une simulation numérique directe d’un écoulement turbulent MHD. Les propriétés de localité du transport de l’énergie entre les échelles pour un écoulement anisotropique ou isotropique, généré par la présence d’un champ magnétique constant, sont renforcées. Un objectif secondaire est d’établir un cadre de travail pour l’étude du transport de particules test chargées dans un champ électromagnétique turbu-lent, i.e. généré par le mouvement d’un fluide conducteur, qui possède des structures à plusieurs ordres de grandeur. La structure de la thèse est présentée ci-dessous.<p>Dans la première partie, composée des deux premiers chapitres, l’auteur présente les notions de turbu-lences, aussi bien hydrodynamiques que MHD. Ces deux chapitres sont des synthèses.<p>La deuxième partie est la principale source de nouveaux résultats. Le chapitre 3 présente les méthodes numériques pour la résolution des équations, les méthodes pseudo-spectrales. Un nouveau type de force est introduit, imposant un niveau de dissipation pour tous les invariants. Dans le chapitre 4, il est effectué une analyse du transfert d'énergie entre ordres de grandeur pour les turbulences MHD. Pour explorer ces transferts d'énergie, le domaine spectral est décomposé en une série de coques de même nombre d'onde. Le transfert moyen d'énergie entre ces coques est analysé. Les transferts d'énergie s'avèrent être surtout locaux en ordre de grandeur, alors qu'une contribution non locale existe due à la force. En présence d'un champ magnétique, l'écoulement développe une direction préférentielle, une anisotropie, où une idée nouvelle de décomposition de l'espace spectral en structures annulaires est présentée. Utilisant cette décomposition annulaire on trouve que le transfert entre anneaux est local, surtout dans les anneaux de direction perpendiculaire au champ magnétique. Pour les turbulences isotropiques, dans le chapitre 5, la localité des flux d'énergie est explorée par le biais de fonctions de localité. Dans le cas de la turbulence MHD, nous avons un comportement non local plus prononcé.<p>La dernière partie, les chapitres 6 et 7, présente le formalisme de suivi des trajectoires de particules chargées évoluant dans un champ électromagnétique turbulent. L'influence de la méthode d'interpola-tion du solveur de particules est étudiée avant la présentation des concepts liés au transport de particu-les et aux régimes de diffusion. L'adiabatisme du mouvement des particules chargées est discuté et le transport de particules chargées dans un champ magnétique turbulent est montré en exemple.<p> / Doctorat en sciences, Spécialisation physique / info:eu-repo/semantics/nonPublished

The context of this thesis is the numerical simulation of turbulent flows at moderate Reynolds numbers and the improvement of the capabilities of an in-house 3D unsteady and incompressible flow solver called SFELES to simulate such flows.<p>In addition to this abstract, this thesis includes five other chapters.<p><p>The second chapter of this thesis presents the numerical methods implemented in the two CFD solvers used as part of this work, namely SFELES and PHASTA.<p><p>The third chapter concentrates on the implementation of a new library called FlexMG. This library allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods, which require much less memory to solve linear systems than a direct sparse LU solver available in SFELES. Multigrid is an iterative procedure that relies on a series of increasingly coarser approximations of the original 'fine' problem. The underlying concept is the following: low wavenumber errors on fine grids become high wavenumber errors on coarser levels, which can be effectively removed by applying fixed-point methods on coarser levels.<p>Two families of algebraic multigrid preconditioners have been implemented in FlexMG, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES (Generalized Minimal RESidual) method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics (unsteady flow past a circular cylinder at low Reynolds number), their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in this work.<p><p>The fourth chapter is devoted to the study of subgrid scale models for the large eddy simulation (LES) of turbulent flows.<p>It is well known that turbulence features a cascade process by which kinetic energy is transferred from the large turbulent scales to the smaller ones. Below a certain size, the smallest structures are dissipated into heat because of the effect of the viscous term in the Navier-Stokes equations.<p>In the classical formulation of LES models, all the resolved scales are used to model the contribution of the unresolved scales. However, most of the energy exchanges between scales are local, which means that the energy of the unresolved scales derives mainly from the energy of the small resolved scales.<p>In this fourth chapter, constant-coefficient-based Smagorinsky and WALE models are considered under different formulations. This includes a classical version of both the Smagorinsky and WALE models and several scale-separation formulations, where the resolved velocity field is filtered in order to separate the small turbulent scales from the large ones. From this separation of turbulent scales, the strain rate tensor and/or the eddy viscosity of the subgrid scale model is computed from the small resolved scales only. One important advantage of these scale-separation models is that the dissipation they introduce through their subgrid scale stress tensor is better controlled compared to their classical version, where all the scales are taken into account without any filtering. More precisely, the filtering operator (based on a top hat filter in this work) allows the decomposition u' = u - ubar, where u is the resolved velocity field (large and small resolved scales), ubar is the filtered velocity field (large resolved scales) and u' is the small resolved scales field. <p>At last, two variational multiscale (VMS) methods are also considered.<p>The philosophy of the variational multiscale methods differs significantly from the philosophy of the scale-separation models. Concretely, the discrete Navier-Stokes equations have to be projected into two disjoint spaces so that a set of equations characterizes the evolution of the large resolved scales of the flow, whereas another set governs the small resolved scales. <p>Once the Navier-Stokes equations have been projected into these two spaces associated with the large and small scales respectively, the variational multiscale method consists in adding an eddy viscosity model to the small scales equations only, leaving the large scales equations unchanged. This projection is obvious in the case of a full spectral discretization of the Navier-Stokes equations, where the evolution of the large and small scales is governed by the equations associated with the low and high wavenumber modes respectively. This projection is more complex to achieve in the context of a finite element discretization. <p>For that purpose, two variational multiscale concepts are examined in this work.<p>The first projector is based on the construction of aggregates, whereas the second projector relies on the implementation of hierarchical linear basis functions.<p>In order to gain some experience in the field of LES modeling, some of the above-mentioned models were implemented first in another code called PHASTA and presented along with SFELES in the second chapter.<p>Finally, the relevance of our models is assessed with the large eddy simulation of a fully developed turbulent channel flow at a low Reynolds number under statistical equilibrium. In addition to the analysis of the mean eddy viscosity computed for all our LES models, comparisons in terms of shear stress, root mean square velocity fluctuation and mean velocity are performed with a fully resolved direct numerical simulation as a reference.<p><p>The fifth chapter of the thesis focuses on the numerical simulation of the 3D turbulent flow over a re-entry Apollo-type capsule at low speed with SFELES. The Reynolds number based on the heat shield is set to Re=10^4 and the angle of attack is set to 180º, that is the heat shield facing the free stream. Only the final stage of the flight is considered in this work, before the splashdown or the landing, so that the incompressibility hypothesis in SFELES is still valid.<p>Two LES models are considered in this chapter, namely a classical and a scale-separation version of the WALE model. Although the capsule geometry is axisymmetric, the flow field in its wake is not and induces unsteady forces and moments acting on the capsule. The characterization of the phenomena occurring in the wake of the capsule and the determination of their main frequencies are essential to ensure the static and dynamic stability during the final stage of the flight. <p>Visualizations by means of 3D isosurfaces and 2D slices of the Q-criterion and the vorticity field confirm the presence of a large meandering recirculation zone characterized by a low Strouhal number, that is St≈0.15.<p>Due to the detachment of the flow at the shoulder of the capsule, a resulting annular shear layer appears. This shear layer is then affected by some Kelvin-Helmholtz instabilities and ends up rolling up, leading to the formation of vortex rings characterized by a high frequency. This vortex shedding depends on the Reynolds number so that a Strouhal number St≈3 is detected at Re=10^4.<p>Finally, the analysis of the force and moment coefficients reveals the existence of a lateral force perpendicular to the streamwise direction in the case of the scale-separation WALE model, which suggests that the wake of the capsule may have some <p>preferential orientations during the vortex shedding. In the case of the classical version of the WALE model, no lateral force has been observed so far so that the mean flow is thought to be still axisymmetric after 100 units of non-dimensional physical time.<p><p>Finally, the last chapter of this work recalls the main conclusions drawn from the previous chapters. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished