Detailed sensitivity numerical studies have shown that the mesh cell-size may have a drastic effect on the modelling of circulating fluidized bed with small particles. Typically, the cell-size must be of the order of few particle diameters to predict accurately the dynamical behaviour of a fluidized bed. Hence, the Euler-Euler numerical simulations of industrial processes are generally performed with grids too coarse to allow the prediction of the local segregation effects. Appropriate modelling, which takes into account the influence of unresolved structures, have been already proposed for monodisperse simulations. In this work, the influence of unresolved structures on a binary mixture of particles is investigated and models are proposed to account for those effect on bidisperse simulations of bidisperse gas-solid fluidized bed. To achieve this goal, Euler-Euler reference simulations are performed with grid refinement up to reach a mesh independent solution. Such kind of numerical simulation is very expensive and is restricted to very simple configurations. In this work, the configuration consists of a 3D periodical circulating fluidized bed, that could represent the established zone of an industrial circulating fluidized bed. In parallel, a filtered approach is developed where the unknown terms, called sub-grid contributions, appear. They correspond to the difference between filtered terms, which are calculated with the reference results then filtered, and resolved contributions, calculated with the filtered fields. Then spatial filters can be applied to reference simulation results to measure each sub-grid contribution appearing in the theoretical filtered approach. A budget analysis is carried out to understand and model the sub-grid term. The analysis of the filtered momentum equation shows that the resolved fluid-particle drag and inter-particle collision are overestimating the momentum transfer effects. The analysis of the budget of the filtered random kinetic energy shows that the resolved production by the mean shear and by the mean particle relative motion are underestimating the filtered ones. Functional models are proposed for the subgrid contributions of the drag and the inter-particle collision.

Des études numériques ont montré que la taille de la cellule de maillage peut avoir un effet drastique sur la modélisation du lit fluidisé circulant avec des petites particules. En effet, la taille de la cellule doit être de l’ordre de quelques diamètres de particules pour prédire avec précision le comportement dynamique d’un lit fluidisé. En conséquence, les simulations numériques d’ Euler-Euler des processus industriels sont généralement effectuées avec des grilles trop grossières pour permettre la prédiction des effets de ségrégation locale. La modélisation appropriée, qui prend en compte l’influence des structures non résolues, a déjà été proposée pour les simulations monodispersés. Dans ce travail, l’influence des structures non résolues sur un mélange binaire de particules est analysée et on propose des modèles pour tenir compte de cet effet dans des simulations de lit fluidisé polydispersés. Pour atteindre cet objectif, des simulations Euler-Euler de références sont réalisées avec un raffinement du maillage aboutissant à une solution indépendante de la taille de la cellule. Ce type de simulation numérique est très coûteux et se limite à des configurations très simples. Dans ce travail, la configuration se consiste en un lit circulant périodique 3D, qui représente la région établie d'un lit circulant. Parallèlement, une approche filtrée est développée où les termes inconnus, appelés contributions de sous-maille, doivent être modélisés. Les filtres spatiaux peuvent être appliqués aux résultats de simulation de référence afin de mesurer chaque contribution de sous-maille apparaissant dans l’approche théorique filtrée. Une analyse est réalisée afin de comprendre et de modéliser l’effet de la contribution des termes de sous-maille. L’opération de filtrage fait apparaître de nouveaux termes, les termes de sous-maille. Un terme filtré est la somme d’un terme résolu, obtenus à partir des champs filtrés, et d’ un terme de sous-maille. L’analyse de l’équation filtrée de quantité de mouvement montre que les contributions résolues de la traînée des particules fluides et la collision entre particules surestiment les effets de transferts de quantité de mouvement filtrés. L’analyse de l’équation filtrée de l’énergie cinétique des particules montre que la production résolue par le cisaillement moyen et par le mouvement relatif moyen des particules sous-estime contribution filtrée. Des modèles fonctionnels sont proposés pour les contributions de sous-maille de la traînée et des collisions inter-particule. / Detailed sensitivity numerical studies have shown that the mesh cell-size may have a drastic effect on the modelling of circulating fluidized bed with small particles. Typically, the cell-size must be of the order of few particle diameters to predict accurately the dynamical behaviour of a fluidized bed. Hence, the Euler-Euler numerical simulations of industrial processes are generally performed with grids too coarse to allow the prediction of the local segregation effects. Appropriate modelling, which takes into account the influence of unresolved structures, have been already proposed for monodisperse simulations. In this work, the influence of unresolved structures on a binary mixture of particles is investigated and models are proposed to account for those effect on bidisperse simulations of bidisperse gas-solid fluidized bed. To achieve this goal, Euler-Euler reference simulations are performed with grid refinement up to reach a mesh independent solution. Such kind of numerical simulation is very expensive and is restricted to very simple configurations. In this work, the configuration consists of a 3D periodical circulating fluidized bed, that could represent the established zone of an industrial circulating fluidized bed. In parallel, a filtered approach is developed where the unknown terms, called sub-grid contributions, appear. They correspond to the difference between filtered terms, which are calculated with the reference results then filtered, and resolved contributions, calculated with the filtered fields. Then spatial filters can be applied to reference simulation results to measure each sub-grid contribution appearing in the theoretical filtered approach. A budget analysis is carried out to understand and model the sub-grid term. The analysis of the filtered momentum equation shows that the resolved fluid-particle drag and inter-particle collision are overestimating the momentum transfer effects. The analysis of the budget of the filtered random kinetic energy shows that the resolved production by the mean shear and by the mean particle relative motion are underestimating the filtered ones. Functional models are proposed for the subgrid contributions of the drag and the inter-particle collision.

Most flows in nature and engineering are turbulent, and many are wall-bounded. Further, in turbulent flows, the turbulence generally has a large impact on the behavior of the flow. It is therefore important to be able to predict the effects of turbulence in such flows. The Navier-Stokes equations are known to be an excellent model of the turbulence phenomenon. In simple geometries and low Reynolds numbers, very accurate numerical solutions of the Navier-Stokes equations (direct numerical simulation, or DNS) have been used to study the details of turbulent flows. However, DNS of high Reynolds number turbulent flows in complex geometries is impractical because of the escalation of computational cost with Reynolds number, due to the increasing range of spatial and temporal scales. In Large Eddy Simulation (LES), only the large-scale turbulence is simulated, while the effects of the small scales are modeled (subgrid models). LES therefore reduces computational expense, allowing flows of higher Reynolds number and more complexity to be simulated. However, this is at the cost of the subgrid modeling problem. The goal of the current research is then to develop new subgrid models consistent with the statistical properties of turbulence. The modeling approach pursued here is that of "Optimal LES". Optimal LES is a framework for constructing models with minimum error relative to an ideal LES model. The multi-point statistics used as input to the optimal LES procedure can be gathered from DNS of the same flow. However, for an optimal LES to be truly predictive, we must free ourselves from dependence on existing DNS data. We have done this by obtaining the required statistics from theoretical models which we have developed. We derived a theoretical model for the three-point third-order velocity correlation for homogeneous, isotropic turbulence in the inertial range. This model is shown be a good representation of DNS data, and it is used to construct optimal quadratic subgrid models for LES of forced isotropic turbulence with results which agree well with theory and DNS. The model can also be filtered to determine the filtered two-point third-order correlation, which describes energy transfer among filtered (large) scales in LES. LES of wall-bounded flows with unresolved wall layers commonly exhibit good prediction of mean velocities and significant over-prediction of streamwise component energies in the near-wall region. We developed improved models for the nonlinear term in the filtered Navier-Stokes equation which result in better predicted streamwise component energies. These models involve (1) Reynolds decomposition of the nonlinear term and (2) evaluation of the pressure term, which removes the divergent part of the nonlinear models. These considerations significantly improved the performance of our optimal models, and we expect them to apply to other subgrid models as well. / text