Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

A Smoothed Particle Hydrodynamics Method for the Simulation of Centralized Sloshing Experiments

The Smoothed Particle Hydrodynamics (SPH) method is proposed for studying hydrodynamic processes related to nuclear engineering problems. A problem of possible recriticality due to the sloshing motions of the molten reactor core is studied with SPH method. The accuracy of the numerical solution obtained in this study with the SPH method is significantly higher than that obtained with the SIMMER-III/IV reactor safety analysis code

Simulating unsteady conduit flows with smoothed particle hydrodynamics

Pipelines are widely used for transport and cooling in industries such as oil and gas, chemical, water supply and sewerage, and hydro, fossil-fuel and nuclear power plants. Unsteady pipe flows with large pressure variations may cause a range of problems such as pipe rapture, support failure, pipe movement, vibration and noise. The unsteady flow is generally caused by flow velocity changes due to valve or pump operation. Water hammer is the best known and extensively studied phenomenon in this respect. Fast transient may also occur in rapid pipe filling and emptying processes. Due to high driving heads, the advancing liquid column may achieve a high velocity. When this high-velocity column is blocked or restricted in its flow, high water-hammer pressures may result. Another scenario is that of slug flow, which arguably is the most dangerous type of two-phase pipe flow. Heavy isolated liquid slugs travelling at high speed behave like cannonballs. Damage is likely to happen when these slugs impact on barriers such as pumps, bends and partially closed valves. Advancing liquid columns occurring in rapid pipe filling and emptying can be seen as a special case of isolated slugs. In this thesis, we present a Lagrangian particle method for solving the Euler equations with application to water hammer, rapid pipe filling and emptying, and isolated slugs travelling in an empty pipeline. As a meshfree method, the smoothed particle hydrodynamics (SPH) used herein is suitable for problems encompassing moving boundaries and impact events, which are the common features of the concerned topics. We first present the kernel and particle approximation concepts, which are two essential steps in SPH. Based on numerical approximation rules, the SPH discrete form of the Euler and Navier-Stokes equations are derived. To treat various boundary conditions, we apply several types of image particles that are particularly designed to complete the kernels truncated by system boundaries. The global conservation of mass and linear momentum is then demonstrated. The SPH errors in the integral approximation and summation approximation are analyzed based on given particle distribution patterns. Several other problems such as particle clustering, tensile instability, particle boundary layer and lacking of polynomial reproducing abilities (incompleteness) are also discussed together with possible remedies. Before applying the implemented particle solver to the thesis topics, we first thoroughly test it against a selection of two-dimensionale benchmarks, which have close relationship with the concerned problems. They include dam-break, jet impinging onto an inclined plane, emerging jet under gravity, free overfall and flow separation at bends. Good agreements with analytical and numerical solutions in literature are found. The convergence rate of SPH is shown to be of first order, which is consistent with the theoretical analysis. For the rapid pipe filling problem, we apply the 1D SPH solver to the experiment of Liou & Hunt [114]. The velocity head at the inlet has to be taken into account to obtain a good agreement with the experiment. Water elasticity does not play a role and the friction formulation for steady state flows can be used. Head transition analysis provides deeper insight into the hydrodynamic behaviour in the filling process. As a special case of pipe filling, water hammer due to liquid impact at partially and fully closed valves is studied. The results agree well with standard MOC solutions. Similar observations are made for the rapid emptying process. For the isolated slug travelling in a voided pipeline and impacting on a bend, we apply the 1D and 2D SPH solvers to the experiments of Bozkus [24]. To obtain the arrival velocity of the slug at the elbow, a 1D model including mass loss at the slug tail is used. In the slug impact, flow separation at the bend plays a vital role, which is typical 2D flow behaviour at a geometrical discontinuity. With the flow contraction coefficient obtained from 2D SPH solutions, the improved 1D model gives good results for the reaction force, not only in magnitude but also its duration and shape. Finally, to study the evolutions of air/water interface and its possible effect on filling and emptying processes, a new experimental study is performed in a large-scale pipeline. It is found that in filling the water front tends to split into two fronts propagating with different velocities. This results in air intrusion on top of a water platform. In emptying, flow stratification occurs at the water tail. Consequently, the validated assumption of vertical air/water interfaces for small-scale system with high driving head may not be applicable to large-scale systems. The interface evolution does not play an important role in pipe filling, the overall behaviour of which can be well predicted with 1D SPH solutions. However, flow stratification largely affects the overall draining process

A Smoothed Particle Hydrodynamics Method for the Simulation of Centralized Sloshing Experiments

The Smoothed Particle Hydrodynamics (SPH) method is proposed for studying hydrodynamic processes related to nuclear engineering problems. A problem of possible recriticality due to the sloshing motions of the molten reactor core has been studied. The accuracy of the numerical solution obtained in this study with the SPH method is significantly higher than that obtained with the SIMMER-III/IV reactor safety analysis code

Red Blood Cell Dynamics on Non-Uniform Grids using a Lattice Boltzmann Flux Solver and a Spring-Particle Red Blood Cell Model

The Computational Haemodynamics Research Group (CHRG) in Technological University Dublin is developing a computational fluid dynamics (CFD) software package aimed specifically at physiologically-realistic modelling of blood flow. A physiologically-realistic model of blood flow involves calculating the deformation of individual red blood cells (RBCs) and the contribution of this deformation to the overall blood flow. The CHRG has developed an enhanced spring-particle RBC structural model that is capable of modelling the full stomatocyte-discocyteechinocyte (SDE) transformation. This RBC model, incorporated into a fluid dynamics solver, will provide a physiologically-realistic blood flow model. In this work the overall plasma flow is modelled using a novel technique: the lattice Boltzmann flux solver (LBFS). This is an innovative approach to solving the NavierStokes (N-S) equations for fluid flow. It involves solving the macroscopic equations using the finite volume method (FVM) and calculating the flux across the cell interfaces via a local reconstruction of the lattice Boltzmann equation (LBE). Fluidstruture interaction between the RBC and the plasma is captured by coupling the RBC solver to the LBFS via the immersed boundary method (IBM). Numerical experiments investigating RBC dynamics are performed using non-uniform grids and validated against existing experimental data in the literature. Finally all numerical solvers are developed using general purpose GPU programming (GPGPU) and this is shown to accelerate simulation runtimes significantly

Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

The use of meshfree and particle methods in the field of bioengineering and biomechanics has significantly increased. This may be attributed to their unique abilities to overcome most of the inherent limitations of mesh-based methods in dealing with problems involving large deformation and complex geometry that are common in bioengineering and computational biomechanics in particular. This review article is intended to identify, highlight and summarize research works on topics that are of substantial interest in the field of computational biomechanics in which meshfree or particle methods have been employed for analysis, simulation or/and modeling of biological systems such as soft matters, cells, biological soft and hard tissues and organs. We also anticipate that this review will serve as a useful resource and guide to researchers who intend to extend their work into these research areas. This review article includes 333 references

Discontinuous Galerkin Spectral Element Methods for Astrophysical Flows in Multi-physics Applications

In engineering applications, discontinuous Galerkin methods (DG) have been proven to be a powerful and flexible class of high order methods for problems in computational fluid dynamics. However, the potential benefits of DG for applications in astrophysical contexts is still relatively unexplored in its entirety. To this day, a decent number of studies surveying DG for astrophysical flows have been conducted. But the adoption of DG by the astrophysics community is just beginning to gain traction and integration of DG into established, multi-physics simulation frameworks for comprehensive astrophysical modeling is still lacking. It is our firm believe, that the full potential of novel approaches for numerically solving the fluid equations only shows under the pressure of real-world simulations with all aspects of multi-physics, challenging flow configurations, resolution and runtime constraints, and efficiency metrics on high-performance systems involved. Thus, we see the pressing need to propel DG from the well-trodden path of cataloguing test results under "optimal" laboratory conditions towards the harsh and unforgiving environment of large-scale astrophysics simulations. Consequently, the core of this work is the development and deployment of a robust DG scheme solving the ideal magneto-hydrodynamics equations with multiple species on three-dimensional Cartesian grids with adaptive mesh refinement. We chose to implement DG within the venerable simulation framework FLASH, with a specific focus on multi-physics problems in astrophysics. This entails modifications of the vanilla DG scheme to make it fit seamlessly within FLASH in such a way that all other physics modules can be naturally coupled without additional implementation overhead. A key ingredient is that our DG scheme uses mean value data organized into blocks - the central data structure in FLASH. Having the opportunity to work on mean values, allows us to rely on a rock-solid, monotone Finite Volume (FV) scheme as "backup" whenever the high order DG method fails in cases when the flow gets too harsh. Finding ways to combine the two schemes in a fail-safe manner without loosing primary conservation while still maintaining high order accuracy for smooth, well-resolved flows involves a series of careful considerations, which we document in this thesis. The result of our work is a novel shock capturing scheme - a hybrid between FV and DG - with smooth transitions between low and high order fluxes according to solution smoothness estimators. We present extensive validations and test cases, specifically its interaction with multi-physics modules in FLASH such as (self-)gravity and radiative transfer. We also investigate the benefits and pitfalls of integrating end-to-end entropy stability into our numerical scheme, with special focus on highly compressible turbulent flows and shocks. Our implementation of DG in FLASH allows us to conduct preliminary yet comprehensive astrophysics simulations proving that our new solver is ready for assessments and investigations by the astrophysics community

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Développement d'un modèle Euler-Lagrange robuste pour la simulation des écoulements solide-liquide dans les opérations de mélange

Les opérations de mélange solide-liquide en cuves agitées jouent un rôle clef dans de nombreux procédés, que ce soit dans la fabrication de produits alimentaires, pharmaceutiques, cosmétiques, ou pour promouvoir l’homogénéité de suspensions, ce qui est particulièrement vital au bon fonctionnement des réacteurs chimiques employant un catalyseur solide. Malgré leur importance indéniable pour l’industrie chimique et les efforts considérables qui ont été déployés afin de mieux les comprendre, la conception et l’optimisation de ces opérations demeurent un grand défi. En effet, la quasi-totalité de la littérature se concentre sur le régime d’opération pleinement turbulent, malgré le fait que de nombreux procédés industriels soient opérés en régimes laminaire ou transitoire. Malheureusement, la littérature sur ces derniers régimes d’opération est quasi-inexistante. De plus, bien que le régime d’opération turbulent ait fait l’objet d’un grand nombre d’études, celles-ci se sont principalement concentrées sur la prédiction, à l’aide de corrélations empiriques ou semi-empiriques, de la vitesse nécessaire pour la suspension complète des particules (Njs), c’est-à dire la vitesse d’agitation nécessaire pour suspendre toutes les particules hors du fond de la cuve. Cependant, de nombreux procédés pourraient être opérés dans des conditions différentes telles que la suspension partielle ou complètement homogène. Le premier cas permettrait d’économiser de l’énergie et d’éviter des contraintes trop fortes sur l’agitateur, tandis que le second assurerait une cinétique de réaction et une qualité de produit nettement mieux contrôlée. Dans ces deux situations, la connaissance de Njs n’est que d’une aide limitée. Pour mieux opérer et concevoir ces unités, il est nécessaire de pouvoir prédire la distribution et la dispersion des particules solides ainsi que les patrons d’écoulement au sein de la cuve. L’étude des systèmes de mélange solide-liquide représente une difficulté importante, car l’opacité des suspensions solide-liquide limite fortement la mesure de variables locales telles que les profils de concentration. Ainsi, la majeure partie des travaux expérimentaux n’ont mesuré que des paramètres globaux, tels que le couple sur l’agitateur, la fraction de particules suspendues ou Njs. La mécanique des fluides numérique (CFD), complémentaire à l’expérience, permet quant à elle d’investiguer à la fois les paramètres globaux, mais aussi ce qui se passe localement en tout point de la cuve. Cependant, la modélisation d’écoulements multiphasiques renferme de nombreux défis compte tenu de l’interaction multiéchelle (de temps et d’espace) entre les phases. Les nombreux modèles capables de modéliser ces types d’écoulements sont décrits dans cette thèse, dans l’objectif de faire ressortir leurs forces ainsi que leurs limites. Parmi ceux-ci, il est montré que seuls les modèles à deux fluides, où la phase solide est modélisée comme un second fluide, ont été utilisés exhaustivement pour aborder le mélange solide-liquide. Cependant, ces modèles souffrent d’une incapacité à bien décrire les régimes rapides et denses d’écoulement granulaire (comportement de Burnett et de super Burnett) ainsi que de plusieurs difficultés à saturer la concentration de solide lorsque les particules sont à leur fraction maximale d’empilement. La CFD-DEM, une famille de modèles relativement récents qui combinent la CFD pour la phase fluide et la méthode des éléments discrets (DEM) pour les particules solides permet quant à elle de décrire la dynamique de la phase granulaire avec un grand degré de précision et a largement fait ses preuves dans l’étude de milieux solide-gaz. Cependant, cette méthode n’a jamais été employée rigoureusement pour l’étude d’écoulement solide-liquide dans des géométries complexes telles que des mélangeurs. Pour que ceci soit possible, de nombreux développements mathématiques sont nécessaires afin de s’assurer que le schéma soit stable, qu’il converge (en temps et en espace) et qu’il soit capable de simuler des géométries complexes en mouvement tels que les agitateurs. La conception d’un tel modèle et son application à l’étude de la dynamique du mélange solide-liquide et de la mise en suspension de particules solides est l’objectif principal de cette thèse. En premier lieu, un schéma volume fini de type Pressure Implicit with Splitting of Operator (PISO) pour résoudre les équations de Navier-Stokes moyennées volumiquement (VANS), nommé PISO-VANS, est établi. La résolution de ces équations est une partie essentielle d’un modèle CFD-DEM applicable à des écoulements concentrés. Afin de vérifier la cohérence du schéma PISO-VANS pour la résolution des équations VANS, une méthodologie basée sur la méthode des solutions manufacturées est développée afin d’établir des cas tests analytiques permettant d’effectuer des tests de convergence numérique. Ces tests, les premiers de ce genre, démontrent que le schéma proposé converge à la précision désirée, c’est-à-dire qu’il est bien de second ordre en temps et en espace. Ensuite, cette méthodologie est employée pour vérifier un nouveau schéma permettant de résoudre les équations VANS avec la méthode de Boltzmann sur réseau (LBM). Ce schéma est basé sur un nouvel opérateur de collision. Il est démontré, à l’aide d’une analyse de Chapmann-Enskogg, que la formulation proposée permet de retrouver les équations VANS. Cet opérateur est le premier permettant de résoudre les équations VANS avec la LBM lorsque la fraction volumique n’est pas constante dans l’espace, une capacité essentielle pour l’étude d’écoulements polyphasiques ou dans des milieux poreux. Dans la troisième partie de ce travail, une nouvelle méthode de condition immergée semiimplicite permettant de modéliser des corps rigides en rotation est développée. Cette méthode est conçue pour bien s’harmoniser avec le schéma PISO et pour être fonctionnelle sur un maillage non structuré polyédrique tout en demeurant parallèle. Cette méthode est tout d’abord vérifiée sur des cas tests académiques tels que l’allée de von Karman derrière un cylindre, ainsi qu’un écoulement de Taylor-Couette entre deux cylindres. Il est montré que le schéma peut bien reproduire les vortex de von Karman, mais qu’il dégrade l’ordre du schéma volume fini de 2 à 1.33 dans le cas de l’écoulement de Taylor-Couette. Le schéma est finalement validé expérimentalement et comparé à d’autres méthodes numériques permettant de simuler des géométries en rotation. Un accord quasi parfait est obtenu. La quatrième partie de ce travail résulte directement de la combinaison du schéma volume fini PISO-VANS avec la méthode de conditions immergées afin de simuler le mélange solideliquide, du démarrage au régime permanent, à l’aide de la CFD-DEM. Différentes stratégies de couplage entre les phases sont testées et il est montré que, contrairement au cas gaz-solide, un couplage explicite est préférable, car il atténue les erreurs de moyenne volumique. Il est aussi démontré qu’un modèle de rhéologie est nécessaire afin de considérer la dissipation à l’échelle inférieure à une taille de maille. Le modèle complet est ensuite validé qualitativement, en comparant des profiles particuliers d’écoulements obtenus au début de la suspension des particules, et quantitativement, à travers la comparaison avec l’expérience de la fraction de particules suspendues mesurée par la technique de pression de jauge (PGT). À nouveau, un excellent accord est obtenu entre les données expérimentales et les résultats du modèle. Dans la cinquième partie de ce travail, le modèle CFD-DEM est modifié afin de permettre l’étude des écoulements turbulents à l’aide de la simulation aux grandes échelles turbulentes (LES). Suite à une validation avec l’expérience, deux nouvelles techniques de mesure de la fraction de particules suspendues, l’analyse lagrangienne de fraction suspendue (LSFA) et l’analyse de fraction de décorrélation (DFA), sont introduites. Les résultats issus de ces méthodes sont ensuite comparés à ceux obtenus numériquement et expérimentalement par la méthode de pression de jauge. Il est montré que ces deux méthodes sont pratiquement aussi précises que la PGT, mais qu’elles sont aussi plus versatiles, car elles peuvent être appliquées à toutes les géométries et ne nécessitent pas de simulations sur des larges plages de vitesse d’agitation. Dans la sixième partie de ce travail, le modèle CFD-DEM est utilisé pour étudier en détail le mélange solide-liquide en régime laminaire et transitoire. Notamment, l’impact du dégagement de l’agitateur et de la présence de chicanes sur la fraction de particules suspendues et la dynamique du mélange solide-liquide est établi. Il est montré que de réduire le dégagement au fond de l’agitateur permet de prévenir l’apparition de zone mortes à haute vitesse. De surcroît, une étude de sensibilité sur les paramètres de la DEM est effectuée et montre que seule la friction entre les particules joue un rôle significatif sur la dynamique de la phase solide. Finalement, une brève discussion permet de résumer les résultats obtenus et de donner de nombreuses pistes de travaux pouvant faire suite à ce qui fut développé dans le cadre de cette thèse. ---------- Despite the fact that solid-liquid mixing plays a key role in the production of a wide variety of consumer goods such as pastes, paints, cosmetics, propellants, pharmaceuticals, and food products as well as in the operation of chemical reactors with solid catalysts, it still faces considerable challenges. Most research on solid-liquid mixing has focused on the fully turbulent regime of operation even though many industrial operations take place in the laminar or transitional regime. In particular, it is unclear how the rheology of a suspension, particle interactions, and a complex rotating geometry impact flow patterns and particles distribution and dispersion in these regimes. Although more is known about the turbulent regime of operation, most research on this type of regime has been devoted to the prediction of the just-suspended speed (Njs ), which is the impeller speed at which all particles are suspended in the liquid phase. However, numerous mixing operations require a different state of operation. For these processes, operating at Njs can lead to energy overconsumption, product fouling, or inhomogeneous reactions due to the presence of dead zones. Consequently, more information on the velocity patterns and distribution of particles in agitated vessels is required. To shed light on issues related to solid-liquid mixing, numerical and experimental investigations are essential. However, due to the opacity of most viscous suspensions, local measurements of the flow field using optical techniques are highly problematic. Consequently, almost all experimental measurements have been limited to determining the global characteristics of the mixing flow such as the fraction of suspended particles or the torque acting on the impeller. However, CFD simulations of these systems do not suffer from these drawbacks. A variety of models have been developed to simulate solid-liquid flows. These include the classic Eulerian-Eulerian (or two-fluid) model and the combination of the Discrete Element Method (DEM) for the particles and CFD method for the liquid phase (CFD-DEM). Although it possesses enormous potential due to its formulation, notably as regards to its natural capacity to reproduce the maximal packing fraction of solid particles, the ability of the CFD-DEM approach to accurately model solidliquid flows in complex geometries has not yet been proved. In addition, the method has not been validated experimentally for solid-liquid flows. However, this type of model could theoretically allow for a quantitative assessment of flow patterns, particle distributions, and the fraction of suspended particles. In this thesis, a CFD-DEM model is developed to model the suspension of particles in a stirred tank, from start-up to steady state and in all regimes of operation. The model is used to improve our understanding of solid-liquid mixing, notably the issue of predicting the fraction of suspended particles. It is shown to be a quantitative tool that can predict the state and dynamics of a suspension. A methodology is designed to verify a Pressure Implicit with Splitting of Operator (PISO) scheme for the volume-averaged Navier-Stokes (VANS) equations (Article 1). We recall that these equations are essential for the unresolved CFD-DEM method. The methodology, which is based on the method of manufactured solutions, is used to design analytical solutions for the VANS equations for which order of convergence analyses are carried out. The validity of the semi-implicit scheme is established by demonstrating the second-order convergence of the scheme for various complex 2D cases. A novel collision operator for the Lattice Boltzmann Method (LBM) is designed to solve the VANS equations (Article 2). It is demonstrated analytically that this operator solves the VANS equations with second-order accuracy. Numerical test cases designed using the process established in Article 1 were used to confirm these results. The model is able to solve cases where there are large void fraction gradients in the domain. To our knowledge, is the first time that this has been achieved. A semi-implicit immersed boundary method (PISO-IB) is developed to study rotating rigid bodies such as impellers (Article 3). The method is verified using academic test cases, namely the Taylor-Couette flow and the Von Karman vortex street behind a static and a moving cylinder. The scheme accurately reproduces vortex shedding, but degrades the order of convergence of the overall finite volume scheme from 2 to 1.33 in the Taylor-Couette case. The PISO-IB method is validated for single phase mixing, and good agreement is obtained between this method and experimental torque measurements. The methods developed in the first and third sections are then combined to formulate a CFD-DEM scheme for solid-liquid mixing (Article 4). The formulation of the model is analyzed, and two coupling approaches (implicit and explicit) are investigated. Explicit coupling leads to a more stable scheme for viscous fluids. However, due to unresolved hydrodynamic dissipation at the particle scale, a rheology model be introduced into the model. The complete model is validated qualitatively using photographs of the peculiar particle dynamics observed experimentally as well as quantitatively by comparing the fraction of suspended particles measured numerically with experimental data obtained using the pressure gauge technique. The validated CFD-DEM model is extended to the turbulent regime using a large-eddy simulation (LES) approach (Article 5). The model accurately reproduces the fraction of suspended particles measured experimentally. Two new techniques to calculate the fraction of suspended particles, the Lagrangian suspended fraction analysis (LSFA) and the decorrelated fraction analysis (DFA), are developed. These two techniques can be used to calculate the fraction of suspended particles for any vessel bottom, without requiring the simulation of numerous impeller velocities, something that cannot be accomplished using the pressure gauge technique. The entire set of tools described in the previous article is used to study the suspension of solid particles in the laminar and transitional regimes in detail (Articles 6). A parametric study of the model parameters is performed. It shows that only the coefficient of friction plays a role in the solid dynamics. Alternative geometries are also studied by varying the impeller clearance and by adding or removing baffles. These results show that reducing the clearance results in a better distribution of particles and prevents the creation of a dead zone below the impeller. This thesis finishes with a short discussion of the overall capabilities of the model and future research that could arise from it