[pt] A teoria da homogeneização de equações diferenciais tornou-se um campo aberto de pesquisa em diversas áreas das ciências exatas e mostrou-se ser uma poderosa ferramenta para a compreensão do comportamento global de materiais heterogêneos. Apesar de ser conhecido que a dedução da lei de Darcy através das equações de Navier-Stokes já é tema debatido há décadas muitas questões continuam em aberto, principalmente a respeito de condições de contorno mais complexas, casos envolvendo fluxos multifásicos e técnicas de homogeneização numérica. Sabe-se que a lei de Darcy se apresenta sob forma de uma relação linear apenas para um intervalo de gradiente hidráulico e que este intervalo se sobrepõe ao intervalo de fluxo laminar do fluido através dos vazios do solo.Se propõe neste trabalho, então, compreender a perda de linearidade na lei de Darcy, a partir da teoria da homogeneização, modificando e explorando os resultados obtidos anteriormente na literatura. / [en] The theory of homogenization of differential equations has become an open field of research in several areas of the exact sciences and has proved to be a powerful tool for understanding the global behavior of heterogeneous materials. Despite knowing that the deduction of Darcy s law through the Navier-Stokes equations has been debated for decades, many questions remain open, mainly regarding more complex boundary cenditions, cases involving multiphase flows and the numerical homogenization techniques. It is known that Darcy s law is presented in the form of a linear relationship only for a range of hydraulic gradient that overlaps the range of laminar flow of fluids through soil voids. Therefore, it is proposed in this work to understand the loss of linearity in Darcy s law, based on the theory of homogenization, modifying and exploring the limit results obtained in the literature.

Numerical solutions of the Navier-Stokes Equations became more popular in recent decades with increasingly accessible and powerful computational resources. Simulations in reconstructed or artificial pore geometries are often performed to gain insight into microscopic fluid flow structures or are used for upscaling quantities of interest, like hydraulic conductivity. A physically adequate representation of pore-scale flow fields requires analysis of large domains. We solve the incompressible NSE in artificial ordered and random pore-space structures. A simple cubic and face-centred packings of spheres placed in a square duct are analysed. For the fluid flow simulations of random media, packings of spheres, icosahedra, and cubes forming unconsolidated sediments are generated using a rigid body simulation software. The Direct Numerical Simulation method is used for the solution of the NSE implemented in the open-source computational fluid dynamics software OpenFOAM. The influence of the number of spheres in ordered packings, the mesh type, and the mesh resolution is investigated for fluid flow up to Reynolds numbers of 100 based on the spheres' diameter. The random media mesh generation method relies on approximate surface reconstruction. The resulting tetrahedral meshes are then used for steady-state simulations and refined based on an a-posteriori error estimator. The fluid flow simulation results can further be used twofold: 1) They provide homogenized hydro-mechanical properties of the analysed medium for the larger meso and macro groundwater flow simulations. A concept of one-way binding for large-scale simulations is presented. 2) Visualisation: A post-processing image rendering technique was employed in interactive and still image visualisation environments allowing better overview over local fluid flow structures. The ogs FEM code for the solution of large-scale groundwater processes was inspected for computational efficiency. The conclusions drawn from this analysis formed the~basis for the implementation of the~new version of the code---ogs6. The improvements include comparison of linear algebra software realisations and an implementation of optimized memory access patterns in FEM-local assembler part.

It was generally accepted that the rate of formation of cerebrospinal °uid (CSF) is independent of intraventricular pressure [26], until A. Sahar and a host of other scientists challenged this belief. A. Sahar substantiated his belief that the rate of (CSF) formation actually depends on intraventricular pressure, see A. Sahar, 1971 [26]. In this work we show that CSF formation depends on some other factors, including the intraventricular pressure. For the purpose of this study, we used the capillary blood °ow model proposed by K.Boryczko et. al., [5] in which blood °ow in the microvessels was modeled as a two-phase °ow; the solid and the liquid volume phase. CSF is formed from the blood plasma [23] which we assume to be in the liquid volume phase. CSF is a Newtonian °uid [2, 23]. The principles and methods of e®ective area" developed by N. Sauer and R. Maritz [21] for studying the penetration of °uid into permeable walls was used to investigate the ¯ltrate momentum °ux from the intracranial capillary wall through the pia mater and epithelial layer of the choroid plexus into the subarachnoid space. We coupled the dynamic boundary equation with the Navier-Stoke's constitutive equation for incompressible °uid, representing the °uid °ow in the liquid volume phase in the capillary to arrive at our model. / Mathematical sciences / M.Sc.

Thesis (MScEng)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: The hydrodynamic characteristics of three typical ship-like hull sections, in different motions, are numerically investigated using FLUENT, 2009. These simple shapes, namely a v-bottom (triangle) hull, a at-bottom (square) hull and a round-bottom (semi-circle) hull, are investigated in uncoupled heave, sway and roll. The problem is described in two dimensions. A combination of numerical methods and models, found in literature, are used to conduct this investigation. Hull characterisation is achieved through the use of hull mass and damping coe cients. These numerically determined coe cients are compared to experimental work conducted by Vugts (1968). A good correlation between the numerical and experimental results exists for the heave and sway cases. By normalising the coe cients, different hulls are comparable to one another. The numerical models used are validated and veri ed. Roll motion remains largely unsolved for very large angles of roll (in excess of 11°). Different uid ow phenomena occurring around the hull sections have varying degrees of in uence on the motions of a hull. It is found that not one of the turbulence models investigated can be employed to globally solve each type of hull-motion case. Also, forced oscillations in computational simulations require considerably more computational time than free-decay oscillating hull simulations. / AFRIKAANSE OPSOMMING: Die hidrodinamiese karakteristieke van verskillende skeepsrompvorms, in verskeie bewegingswieë, is numeries ondersoek met behulp van FLUENT, 2009. Drie eenvoudige vorms ('n v-bodem (driehoek), plat-bodem (reghoek) en rondebodem (semi-sirkel) romp) is onderskeidelik ondersoek in opwieg, dwarswieg en rol. Die probleem is twee-dimensioneel. Daar is gebruik gemaak van 'n kombinasie van numeriese metodes en modelle, uit die literatuur, om die ondersoek uit te voer. Die rompe is gekarakteriseer met behulp van massa- en dempingskoëffi siënte. Hierdie numeries bepaalde koë ffisiënte is vergelyk met die eksperimentele werk van Vugts (1968). Daar bestaan 'n goeie korrelasie tussen die numeriese en eksperimentele resultate vir die opwieg en dwarswieg gevalle. Die koë ffisiënte is genormaliseer om die verskeie rompvorms te vergelyk. Die numeriese modelle is geverifi eer en valideer. Rolbewegings is onopgelos vir groot rolhoeke (groter as 11°). Die mate waartoe die romp se beweging beïnvloed word deur die verskillende vloei verskynsels wat om die rompe ontstaan, verskil. Daar is bevind dat geen van die turbulensie modelle gebruik kan word om alle skeepsbeweging-gevalle op te los nie. Gedwonge-ossilasie numeriese simulasies benodig meer berekeningstyd as vrye-verval ossilasie gevalle.

Ce travail porte sur la simulation numérique par éléments finis, d'écoulements tridimensionnels incompressibles à surface libre instationnaires. L'application industrielle visée est l'étude de la phase de remplissage des procédés de fonderie. Il consiste principalement en l'adaptation du logiciel REM3D® qui est un logiciel initialement dédié à l'injection des polymères. Les méthodes présentes dans ce code permettent de résoudre les écoulements visqueux à surface libre, dans une approche multidomaine. La résolution directe d'un problème de Navier-Stokes est effectuée en introduisant, dans le solveur initial, les termes de gravité et d'inertie. Ces derniers sont exprimés à l'aide d'une approximation P1 par morceaux de la vitesse, et de la vitesse moyenne par élément, qui est P0 par morceaux : les termes d'advection sont exprimés par une méthode Galerkin discontinu standard. Le problème en vitesse/pression est stabilisé par condensation de bulle. Cette méthode a été testée sur plusieurs cas académiques, pour des écoulements stationnaires et instationnaires, avec suivi de la surface libre. Enfin, afin de pouvoir simuler les écoulements rencontrés en fonderie, des conditions de contact glissant ont été introduites. Un algorithme original de construction de normales multiples conservatives aux nœuds a été implémenté et validé, dans un contexte multidomaine. Le logiciel de simulation ainsi construit, a été validé sur des cas tests spécialement mis au point pour la fonderie, et en thermique multidomaine.

La thèse est consacrée essentiellement à l'étude de systèmes non linéaires d'évolution issus d'un modèle de Boussinesq : couplage entre les équations de Navier-stokes avec un second membre F(µ), où F est une force de gravité proportionnelle à des variations de densité qui dépendent de la température et l'équation de l'énergie.<br />Le premier chapitre nous donne un résultat d'existence d'une solution faible-renormalisée du système de Boussinesq en dimension 2, dans le cas où F est bornée.<br />Dans le chapitre 2, on aborde le cas de fonctions F plus générales : F vérifie une hypothèse de croissance. On démontre l'existence de solutions pour toutes données initiales ou pour des données initiales petites selon la croissance de F.<br />Dans le chapitre 3, nous faisons une généralisation des résultats du chapitre 2 mais sans le terme de convection.<br />Dans le chapitre 4, le manque de stabilité de l'énergie de dissipation dans L1(Q) en dimension 3, nous contraint à transformer de façon formelle le système de Boussinesq. On démontre l'existence d'une solution faible de ce nouveau système en dimension 3.

Two numerical models have been developed to describe the volumetric changes during solidification in pure metals and alloys and to predict shrinkage defects in the castings of general three-dimensional configuration. The first model is based on the full system of the Continuity, Navier-Stokes and Enthalpy Equations. Volumetric changes are described by introducing a source term in the Continuity Equation which is a function of the rate of local phase transformation. The model is capable of simulating both volumetric shrinkage and expansion. The second simplified shrinkage model involves the solution of only the Enthalpy Equation. Simplifying assumptions that the feeding flow is governed only by gravity and solidification rate and that phase transformation proceeds only from liquid to solid allowed the fluid flow equations to be excluded from consideration. The numerical implementation of both models is based on an existing proprietary general purpose CFD code, FLOW-3D, which already contains a numerical algorithm for incompressible fluid flow with heat transfer and phase transformation. An important part of the code is. the Volume Of Fluid (VOF) algorithm for tracking multiple free surfaces. The VOF function is employed in both shrinkage models to describe shrinkage cavity formation. Several modifications to FLOW-3D have been made to improve the accuracy and efficiency of the metal/mould heat transfer and solidification algorithms. As part of the development of the upwind differencing advection algorithm used in the simulations, the Leith's method is incorporated into the public domain twodimensional SOLA code. It is shown that the resulting scheme is unconditionally stable despite being explicit.

La compréhension de l'aérothermique d'un véhicule durant sa phase de développement est une question essentielle afin d'assurer, d'une part, un bon refroidissement et une bonne efficacité de ses composants et d'autre part de réduire la force de traînée et évidement le rejet des gaz à effet de serre ou la consommation d'essence. Cette thèse porte sur la simulation numérique et la validation expérimentale de l'aérothermique d'un véhicule à trois roues dont deux, en avant et une roue motrice en arrière. La simulation numérique est basée sur la résolution des équations de conservation de la masse, de la quantité de mouvement et de l'énergie en utilisant l'approche RANS (Reynolds-Averaged Navier-Stokes). Le rayonnement thermique est modélisé grâce à la méthode S2S (Surface to Surface) qui suppose que le milieu séparant les deux surfaces rayonnantes, ici de l'air, ne participe pas au processus du rayonnement. Les radiateurs sont considérés comme des milieux poreux orthotropes où la perte de pression est calculée en fonction de leurs propriétés inertielle et visqueuse; leur dissipation thermique est modélisée par la méthode Dual flow. Une première validation de l'aérodynamique est faite grâce à des essais en soufflerie. Ensuite, une deuxième validation de la thermique est faite grâce à des essais routiers. Un deuxième objectif de la thèse est consacré à la simulation numérique de l'aérodynamique en régime transitoire du véhicule. La simulation est faite à l'aide de l'approche Detached eddy simulation (DES). Une validation expérimentale est faite à partir d'étude en soufflerie grâce à des mesures locales de vitesse à l'aide de sondes cobra.

In the present work we implemented parallel version of a computational fluid dynamics code. This code is based on Discontinuous Galerkin Method (DGM), which is due to its favourable properties suitable for parallelization. In the work we describe the Navier-Stokes equations and their discretization using DGM. We explain the advantages of usage of the DGM and formulate the serial algorithm. Next we focus on the parallel implementation of the algorithm and several particular issues connected to the parallelization. We present the numerical experiments showing the efficiency of the parallel code in the last chapter.

This thesis is concerned on the simulation of wet steam flow using discontinuous Galerkin method. Wet steam flow equations consist of Naviere-Stokes equations for compressible flow and Hill's equations for condensation of water vapor. The first part of this thesis describes the mathematical formulation of wet steam model and the derivation of Hill's equations. The model equations are discretized with the aid of discontinuous Galerkin method and backward difference formula which leads to implicit scheme represented by nonlinear algebraic system. This system is solved using Newton-like method. The derived scheme was implemented in program ADGFEM which is used for solving non-stationary convective-diffusive problems. The numerical results are presented in the last part of this thesis. 1