Le schéma Marker-And-Cell (MAC) est un schéma de discrétisation des équations aux dérivées partielles sur maillages cartésiens, très connu en mécanique des fluides. Nous nous intéressons ici à son analyse mathématique dans le cadre des écoulements incompressibles sur des maillages cartésiens non-uniformes en dimension 2 ou 3. Dans un premier temps nous discrétisons les équations de Navier-Stokes pour un écoulement incompressible stationnaire; nous établissons des estimations a priori sur les suites de vitesses et pressions approchées qui permettent d’une part d'établir l’existence d’une solution au schéma, et d’obtenir la compacité de ces suites lorsque le pas d’espace tend vers 0. Nous montrons alors la convergence de ces suites (à une sous-suite près) vers une solution faible du problème continu, ce qui nécessite une analyse fine du terme de convection non linéaire. Nous nous intéressons ensuite aux équations de Navier-Stokes en régime instationnaire avec une discrétisation en temps implicite. Nous démontrons que le schéma préserve les propriétés de stabilité du problème continu et obtenons ainsi l’existence d’une solution au schéma. Puis, grâce à des techniques de compacité et en passant à la limite dans le schéma, nous démontrons qu’une suite de vitesses approchées converge. Si l’on se restreint au problème de Stokes, et en supposant de plus que la condition initiale de la vitesse est dans H 1 , nous obtenons une estimation sur la pression qui permet de montrer la convergence forte des pressions approchées. Enfin nous étendons l’analyse aux écoulements incompressibles à masse volumique variable. On montre la convergence du schéma. / The Marker-And-Cell (MAC) scheme is a discretization scheme for partial derivative equations on Cartesian meshes, which is very well known in fluid mechanics. Here we are concerned with its mathematical analysis in the case of incompressible flows on two or three dimensional non-uniform Cartesian grids. We first discretize the steady-state incompressible Navier-Stokes equations. We show somea priori estimates that allow to show the existence of a solution to the scheme and some compactness and consistency results. By a passage to the limit on the scheme, we show that the approximate solutions obtained with the MAC scheme converge (up to a subsequence) to a weak solution of the Navier-Stokes equations, thanks to a careful analysis of the nonlinear convection term. Then, we analyze the convergence of the unsteady-case Navier-Stokes equations. The algorithm is implicit in time. We first show that the scheme preserves the stability properties of the continuous problem, which yields, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem. If we restrict ourselves to the Stokes equations and assume that the initial velocity belongs to H 1, then we obtain estimates on the pressure and prove the convergence of the sequences of approximate pressures. Finally, we extend the analysis of the scheme to incompressible variable density flows. we show the convergence of the scheme.

Cette thèse est dédiée au développement et à l’analyse de schémas numériques de type volumes finis pour des équations de convection-diffusion, qui apparaissent notamment dans des modèles issus de la physique ou de la biologie. Nous nous intéressons plus particulièrement à la préservation de comportements asymptotiques au niveau discret. Ce travail s’articule en trois parties, composées chacune de deux chapitres. Dans la première partie, nous considérons la discrétisation du système de dérive diffusion linéaire pour les semi-conducteurs par le schéma de Scharfetter-Gummel implicite en temps. Nous nous intéressons à la préservation par ce schéma de deux types d’asymptotiques : l’asymptotique en temps long et la limite quasi-neutre. Nous démontrons des estimations d’énergie–dissipation d’énergie discrètes qui permettent de prouver d’une part la convergence en temps long de la solution approchée vers une approximation de l’équilibre thermique, d’autre part la stabilité à la limite quasi-neutre du schéma. Dans la deuxième partie, nous nous intéressons à des schémas volumes finis préservant l’asymptotique en temps long dans un cadre plus général. Plus précisément, nous considérons des équations de type convection-diffusion non linéaires qui apparaissent dans plusieurs contextes physiques : équations des milieux poreux, système de dérive-diffusion pour les semi-conducteurs... Nous proposons deux discrétisations en espace permettant de préserver le comportement en temps long des solutions approchées. Dans un premier temps, nous étendons la définition du flux de Scharfetter-Gummel pour une diffusion non linéaire. Ce schéma fournit des résultats numériques satisfaisants si la diffusion ne dégénère pas. Dans un second temps, nous proposons une discrétisation dans laquelle nous prenons en compte ensemble les termes de convection et de diffusion, en réécrivant le flux sous la forme d’un flux d’advection. Le flux numérique est défini de telle sorte que les états d’équilibre soient préservés, et nous utilisons une méthode de limiteurs de pente pour obtenir un schéma précis à l’ordre deux en espace, même dans le cas dégénéré. Enfin, la troisième et dernière partie est consacrée à l’étude d’un schéma numérique pour un modèle de chimiotactisme avec diffusion croisée pour lequel les solutions n’explosent pas en temps fini, quelles que soient les données initiales. L’étude de la convergence du schéma repose sur une estimation d’entropie discrète nécessitant l’utilisation de versions discrètes d’inégalités fonctionnelles telles que les inégalités de Poincaré-Sobolev et de Gagliardo-Nirenberg-Sobolev. La démonstration de ces inégalités fait l’objet d’un chapitre indépendant dans lequel nous proposons leur étude dans un contexte assez général, incluant notamment le cas de conditions aux limites mixtes et une généralisation au cadre des schémas DDFV. / This dissertation is dedicated to the development and analysis of finite volume numericals chemes for convection-diffusion equations, which notably occur in models arising from physics and biology. We are more particularly interested in preserving asymptotic behavior at the discrete level. This dissertation is composed of three parts, each one including two chapters. In the first part, we consider the discretization of the linear drift-diffusion system for semiconductors with the implicit Scharfetter-Gummel scheme. We focus on preserving two kinds of asymptotics with this scheme : the long-time asymptotic and the quasineutral limit. We show discrete energy–energy dissipation estimates which constitute the main point to prove first the large time convergence of the approximate solution to an approximation of the thermal equilibrium, and then the stability at the quasineutral limit. In the second part, we are interested in designing finite volume schemes which preserve the long time behavior in a more general framework. More precisely, we consider nonlinear convection-diffusion equations arising in various physical models : porous media equation, drift-diffusion system for semiconductors... We propose two spatial discretizations which preserve the long time behavior of the approximate solutions. We first generalize the Scharfetter-Gummel flux for a nonlinear diffusion. This scheme provides satisfying numerical results if the diffusion term does not degenerate. Then we propose a discretization which takes into account together the convection and diffusion terms by rewriting the flux as an advective flux. The numerical flux is then defined in such a way that equilibrium states are preserved, and we use a slope limiters method so as to obtain second order space accuracy, even in the degenerate case. Finally, the third part is devoted to the study of a numerical scheme for a chemotaxis model with cross diffusion, for which the solutions do not blow up in finite time, even for large initial data. The proof of convergence is based on a discrete entropy estimate which requires the use of discrete functional inequalities such as Poincaré-Sobolev and Gagliardo-Nirenberg-Sobolev inequalities. The demonstration of these inequalities is the subject of an independent chapter in which we propose a study in quite a general framework, including mixed boundary conditions and generalization to DDFV schemes.

When an electrically conducting fluid moves through a magnetic field, fluid mechanics and electromagnetism are coupled.<p>This interaction is the object of magnetohydrodynamics, a discipline which covers a wide range of applications, from electromagnetic processing to plasma- and astro-physics.<p><p>In this dissertation, the attention is restricted to turbulent liquid metal flows, typically encountered in steel and aluminium industries. Velocity measurements in such flows are extremely challenging because liquid metals are opaque, hot and often corrosive. Therefore, non-intrusive measurement devices are essential. One of them is the Lorentz force flowmeter. Its working principle is based on the generation of a force acting on a charge, which moves in a magnetic field. Recent studies have demonstrated that this technique can measure efficiently the mean velocity of a liquid metal. In the existing devices, however, the measurement depends on the electrical conductivity of the fluid. <p><p>In this work, a novel version of this technique is developed in order to obtain measurements that are independent of the electrical conductivity. This is particularly appealing for metallurgical applications, where the conductivity often fluctuates in time and space. The study is entirely numerical and uses a flexible computational method, suitable for industrial flows. In this framework, the cost of numerical simulations increases drastically with the level of turbulence and the geometry complexity. Therefore, the simulations are commonly unresolved. Large eddy simulations are then very promising, since they introduce a subgrid model to mimic the dynamics of the unresolved turbulent eddies. <p><p>The first part of this dissertation focuses on the quality and reliability of unresolved numerical simulations. The attention is drawn on the ambiguity that may arise when interpretating the results. Owing to coarse resolutions, numerical errors affect the performances of the discrete model, which in turn looses its physical meaning. In this work, a novel implementation of the turbulent strain rate appearing in the models is proposed. As opposed to its usual discretisation, the present strain rate is in accordance with the discrete equations of motion. Two types of flow are considered: decaying turbulence located far from boundaries, and turbulent flows between two parallel and infinite walls. Particular attention is given to the balance of resolved kinetic energy, in order to assess the role of the model.<p><p>The second part of this dissertation deals with a novel version of Lorentz force flowmeters, consisting in one or two coils placed around a circular pipe. The forces acting on each coil are recorded in time as the liquid metal flows through the pipe. It is highlighted that the auto- or cross-correlation of these forces can be used to determine the flowrate. The reliability of the flowmeter is first investigated with a synthetic velocity profile associated to a single vortex ring, which is convected at a constant speed. This configuration is similar to the movement of a solid rod and enables a simple analysis of the flowmeter. Then, the flowmeter is applied to a realistic three-dimensional turbulent flow. In both cases, the influence of the geometrical parameters of the coils is systematically assessed. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

Ce travail traite de la modélisation des plasmas de fusion qui est ici abordée à l'aide d'un modèle Euler bi-températures et du modèle de la magnétohydrodynamique (MHD) idéale et résistive. Ces modèles sont tout d'abord établis à partir des équations de la MHD bi-fluide et nous montrons qu'ils correspondent à des régimes asymptotiques différents pour des plasmas faiblement ou fortement magnétisés. Nous décrivons ensuite les méthodes de volumes finis pour des maillages structurés et non-structurés qui ont été utilisées pour approcher les solutions de ces modèles. Pour les trois modèles mathématiques étudiés dans cette thèse, les méthodes numériques reposent sur des schémas de relaxation. Afin d'appliquer ces méthodes aux problèmes de fusion par confinement magnétique, nous décrivons comment modifier les méthodes de volumes finis pour les appliquer à des problèmes formulés en coordonnées cylindriques tout en gardant une formulation conservative forte des équations. Enfin nous étudions une stratégie pour maintenir la contrainte de divergence nulle du champ magnétique qui apparait dans les modèles MHD. Une série de cas tests numériques pour les trois modèles est présentée pour différentes géométries afin de valider les méthodes numériques proposées. / This work deals with the modeling of fusion plasma which is discussed by using a bi-temperature Euler model and the ideal and resistive magnetohydrodynamic (MHD) ones. First, these models are established from the bi-fluid MHD equations and we show that they correspond to different asymptotic regimes for lowly or highly magnetized plasma. Next, we describe the finite volume methods for structured and non-structured meshes which have been used to approximate the solution of these models. For the three mathematical models studied in this thesis, the numerical methods are based on relaxation schemes. In order to apply those methods to magnetic confinement fusion problems, we explain how to modify the finite volume methods to apply it to problem given in cylindrical coordinates while keeping a strong conservative formulation. Finally, a strategy dealing with the divergence-free constraint of the magnetic field is studied. A set of numerical tests for the three models is presented for different geometries to validate the proposed numerical methods.

This thesis deals with the development of a new adaptive algorithm for three-dimensional fluid flows based on a residual error estimator. The residual, known as the R –parameter has been successfully extended to three dimensions using a novel approach for arbitrary grid topologies. The computation of the residual error estimator in three dimensions is based on a least-squares based reconstruction and the order of accuracy of the latter is critical in obtaining a consistent estimate of the error. The R –parameter can become inconsistent on three–dimensional meshes depending on the grid quality. A Zero Mean Polynomial(ZMP) which is k–exact, and which preserves the mean has been used in this thesis to overcome the problem. It is demonstrated that the ZMP approach leads to a more accurate estimation of solution derivatives as opposed to the conventional polynomial based least-squares method. The ZMP approach is employed to compute the R –parameter which is the n used to derive the criteria for refinement and derefinement. Studies on three different complex test problems involving inviscid, laminar and turbulent flows demonstrate that the new adaptive algorithm is capable of detecting the sources of error efficiently and lead to accurate results independent of the grid topology.

This Ph.D. thesis aims to develop numerical models for two-dimensional and three-dimensional shallow water systems over mobile beds. In order to accomplish the goal of this dissertation, the following sub-projects are defined and completed. 1: The first sub-project consists in developing a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. In this task, the central-upwind scheme is extended to simulation of bed erosion and sediment transport, a modified shallow water system is adopted to improve the model, a wetting and drying scheme is proposed for tracking wet-dry interfaces and stably predict the bed erosion near wet-dry area. The shallow water, sediment transport and bed evolution equations are coupled in the governing system. The proposed model can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The additional terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate in dealing with various challenging cases. 2: The second sub-project consists in developing a novel numerical scheme for a coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The resulting 5*5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A spatially second-order and temporally third-order central-upwind scheme has been derived to discretize the conservative hyperbolic sub-system. However, such schemes need a correct evaluation of local wave speeds to avoid instabilities. To address such an issue, a mathematical result by the Lagrange theorem is used in the proposed scheme. Consequently, a computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. In addition, a special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. 3: The third sub-project consists in designing a novel scheme to estimate bed-load fluxes which can produce more accurate results than the previously reported coupled model. Using a pair of local wave speeds different from those used for the flow, a novel wave estimator in conjunction with the central upwind method is proposed and successfully applied to the coupled water-sediment system involving a rapid bed-erosion process. It was demonstrated that, in comparison with the decoupled model, applying the proposed novel scheme to approximate the bed-load fluxes can successfully avoid the numerical oscillations caused by simple and less stable schemes, e.g. simple upwind methods; in comparison with the coupled model using same flux-estimator for both hydrodynamic and morphological systems, the proposed numerical scheme successfully prevents excessive numerical diffusion for prediction of bed evolution. Consequently, the proposed scheme has advantages in terms of accuracy which are shown in several numerical tests. In addition, analytical expressions have been provided for calculating the eigenvalues of the coupled shallow-water-Exner system, which greatly enhances the efficiency of the proposed method. 4: The fourth sub-project consists in developing a three-dimensional numerical model for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme which simulates sub- and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a sigma coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term. The unstructured grid in the horizontal direction and the sigma coordinate in the vertical direction facilitate the use of the model in complicated geometries. 5: The fifth sub-project consists in developing a well-balanced three-dimensional shallow water model which is able to simulate shock waves over dry bed. Due to the hydrostatic simplification of the vertical momentum equation, the governing system of equations is not hyperbolic and can not be solved using standard hyperbolic solvers. That is, one can not use a high-order Godunov-type scheme to compute all fluxes through cell-interfaces. This may cause the model to fail in simulations of some unsteady-flows with discontinuities, e.g., dam-break flows and floods. To overcome this difficulty, a novel numerical scheme for the three-dimensional shallow water equations is proposed using a relaxation approach in order to convert the system to a hyperbolic one. Thus, a high-order Godunov-type central-upwind scheme based on the finite volume method can be applied to approximate the numerical fluxes. The proposed model can also preserve the ``lake at rest'' state and positivity of water depth over irregular bottom topographies based on special reconstruction of the corresponding parameters. 6: The sixth sub-project consists in extending the result of the fifth sub-project to development of a three-dimensional numerical model for shallow water flows over mobile beds, which is able to simulate morphological evolutions under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations using a finite volume method on prismatic cells in sigma coordinates based on the scheme prposed in sub-project 5. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. The performance of the proposed model has been demonstrated by several laboratory experiments of dam-break flows over mobile beds.

Nous considérons deux modèles EDP-EDO couplés: un pour modéliser des goulots d’étranglementmobiles et l’autre pour décrire la distribution du trafic sur une bretelle d’accès. Le premier modèle a étéintroduit pour décrire le mouvement d’un bus, qui roule à une vitesse inférieure à celle des autresvoitures, en réduisant la capacité de la route et générant ainsi un goulot d’étranglement. Une loi deconservation scalaire avec une contrainte mobile sur le flux décrit le trafic et une EDO décrit latrajectoire du bus. Nous présentons un résultat d’existence des solutions du modèle et nous proposonsune méthode numérique “front/capturing" et une méthode basée sur une technique de reconstructiondes ondes de chocs. Dans la deuxième partie, nous introduisons un nouveau modèle macroscopique dejonction pour les bretelles d’autoroute. Nous considérons le modèle de trafic de Lighthill-Whitham-Richards sur une jonction composée d’une voie principale, une bretelle d’accès et une bretelle de sortie,toutes reliées par un nœud. Une loi de conservation scalaire décrit l’évolution de la densité des véhiculessur la voie principale et une EDO décrit l’évolution de la longueur de la file d’attente sur la bretelled’accès. La définition de la solution du problème de Riemann à la jonction est basée sur la résolutiond’un problème d’optimisation linéaire et sur l’utilisation d’un paramètre de priorité. Ensuite, ce modèleest étendu aux réseaux et discrétisé en utilisant un schéma de Godunov qui prend en compte les effetsde la bretelle d’accès. Enfin, nous présentons un modèle d’optimisation de la circulation sur les ronds points. / In this thesis we consider two coupled PDE-ODE models. One to model moving bottlenecks and theother one to describe traffic flow at junctions. First, we consider a strongly coupled PDE-ODE systemthat describes the influence of a slow and large vehicle on road traffic. The model consists of a scalarconservation law accounting for the main traffic evolution, while the trajectory of the slower vehicle isgiven by an ODE depending on the downstream traffic density. The moving constraint is expressed byan inequality on the flux, which models the bottleneck created in the road by the presence of the slowerDépôt de thèse – Donnéescomplémentairesvehicle. We prove the existence of solutions to the Cauchy problem for initial data of bounded variation.Moreover, two numerical schemes are proposed. The first one is a finite volume algorithm that uses alocally nonuniform moving mesh. The second one uses a reconstruction technique to display thebehavior of the vehicle. Next, we consider the Lighthill-Whitham-Richards traffic flow model on ajunction composed by one mainline, an onramp and an offramp, which are connected by a node. Theonramp dynamics is modeled using an ordinary differential equation describing the evolution of thequeue length. The definition of the solution of the Riemann problem at the junction is based on anoptimization problem and the use of a right of way parameter. The numerical approximation is carriedout using a Godunov scheme, modified to take into account the effects of the onramp buffer. Aftersuitable modification, the model is used to solve an optimal control problem on roundabouts. Two costfunctionals are numerically optimized with respect to the right of way parameter.

Nous élaborons et analysons mathématiquement des approximations numériques par des méthodes de type volumes finis de solutions faibles de systèmes hyperboliques pour des écoulements géophysiques. Dans une première partie nous approchons les solutions du système de la magnétohydrodynamique en faible épaisseur avec un fond plat. Nous développons un schéma de type Godunov utilisant un solveur de Riemann approché défini via une méthode de relaxation. Des expressions explicites sont établies pour les vitesses de relaxation, qui permettent d'obtenir un schéma satisfaisant un ensemble de bonnes propriétés de consistance et de stabilité. Il conserve la masse, préserve la positivité de la hauteur de fluide, vérifie une inégalité d'entropie discrète, résout les discontinuités de contact même résonantes, donne des vitesses de propagations contrôlées par les données initiales. Des tests numériques sont effectués, validant les résultats théoriques énoncés. Dans une seconde partie nous approchons les solutions du système de la magnétohydrodynamique en faible épaisseur avec fond variable. Nous développons un schéma équilibre pour certains états stationnaires au repos. Nous utilisons la méthode de reconstruction hydrostatique, avec des états reconstruits pour la hauteur d'eau et les composantes du champ magnétique. Nous trouvons des termes correctifs pour les flux numériques par rapport au cadre habituel, et nous prouvons que le schéma obtenu préserve la positivité de la hauteur d'eau, vérifie une inégalité d'entropie semi-discrète et est consistant. Des tests numériques sont effectués, validant les résultats théoriques. Dans une troisième partie nous établissons la convergence d'un schéma cinétique avec reconstruction hydrostatique pour le système de Saint-Venant avec topographie. De nouvelles estimations sur le gradient des solutions approchées sont obtenues par l'analyse de la dissipation d'énergie. La convergence est obtenue par la méthode de compacité par compensation, sous des hypothèses sur les données initiales et la régularité du fond / We build and analyze mathematically numerical approximations by finite volume methods of weak solutions to hyperbolic systems for geophysical flows. In a first part we approximate the solutions of the shallow water magneto hydrodynamics system with flat bottom. We develop a Godunov scheme using an approximate Riemann solver defined via a relaxation method. Explicit formulas are established for the relaxation speeds, that lead to a scheme satisfying good properties of consistency and stability. It preserves mass, positivity of the fluid height, satisfies a discrete entropy inequality, resolves contact discontinuities, and involves propagation speeds controlled by the initial data. Several numerical tests are performed, endorsing the theoretical results. In a second part we approximate the solutions of the shallow water magneto hydrodynamics system with non-flat bottom. We develop a well-balanced scheme for several steady states at rest. We use the hydrostatic reconstruction method, with reconstructed states for the fluid height and the magnetic field. We get some new corrective terms for the numerical fluxes with respect to the classical framework, and we prove that the obtained scheme preserves the positivity of height, satisfies a semi-discrete entropy inequality, and is consistent. Several numerical tests are presented, endorsing the theoretical results. In a third part we prove the convergence of a kinetic scheme with hydrostatic reconstruction for the Saint-Venant system with topography. Some new estimates on the gradient of approximate solutions are established, by the analysis of energy dissipation. The convergence is obtained by the compensated compactness method, under some hypotheses concerning the initial data and the regularity of the topography

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2017-07-14T12:28:59Z No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Dissertacao Bautista Nilson.pdf: 11338318 bytes, checksum: ae22c70eb1719f066a5eeb3de436c953 (MD5) / Made available in DSpace on 2017-07-14T12:28:59Z (GMT). No. of bitstreams: 2 license_rdf: 811 bytes, checksum: e39d27027a6cc9cb039ad269a5db8e34 (MD5) Dissertacao Bautista Nilson.pdf: 11338318 bytes, checksum: ae22c70eb1719f066a5eeb3de436c953 (MD5) Previous issue date: 2016-01-26 / CNPQ / A modelagem e a simulação numérica do transporte de solutos, como por exemplo traçadores, em meios porosos heterogêneos e anisotrópicos, tais como aquíferos e reservatórios de petróleo constituem-se num grande desafio de natureza matemática e numérica. A modelagem de falhas selantes, canais, poços inclinados, pinchouts e outras características complexas demanda o uso de malhas não-estruturadas e não-ortogonais, capazes de se adaptar naturalmente ao domínio em estudo. Os pacotes computacionais utilizados comumente na indústria do petróleo, na sua grande maioria, se baseiam no Método das Diferenças Finitas com Aproximação de Fluxo por Dois Pontos (Two-Point Flux Approximation - TPFA) e no Método de Ponderação à Montante de Primeira Ordem (First Order Upwind Method - FOU), devido a sua facilidade de implementação e sua eficiência computacional. Infelizmente, os métodos TPFA são incapazes de produzir soluções convergentes em malhas não-ortogonais ou para tensores de dispersão ou permeabilidades completos e os métodos FOU produzem soluções com difusão numérica excessiva, exigindo malhas demasiadamente refinadas para obtermos soluções confiáveis. Uma alternativa ao TPFA, e que permite o uso de tensores completos e malhas não-ortogonais, é o Método dos Elementos Finitos de Galerkin (MEF), porém este método não produz soluções localmente conservativas, o que pode ser um problema sério para a modelagem de problemas envolvendo leis de conservação, como no escoamento em meios porosos. Outra alternativa são os Métodos de Volumes Finitos (MVF). Nas suas variantes mais robustas, estes métodos são capazes de lidar com malhas poligonais quaisquer e tensores de dispersão e permeabilidades completos e com razão de anisotropia arbitrária, além de produzir aproximações discretas de alta ordem e localmente conservativas. Neste contexto, no presente trabalho, apresentamos uma formulação MVF centrado na célula para a modelagem do transporte de um traçador não-reativo num escoamento monofásico em meios porosos heterogêneos e anisotrópicos. Para a discretização dos termos elípticos, tanto da equação de pressão quanto da equação de Advecção-Dispersão-Reação (ADRE), utilizou-se um MVF com aproximação de fluxo por múltiplos pontos que faz uso do estêncil diamante (MPFA-D) e para a discretização dos termos hiperbólicos, usamos o método FOU e um MVF do tipo MUSCL (Monotone Upstream Centered Scheme for Conservation Laws). A fim de testar nossa formulação, resolvemos alguns problemas benchmark encontrados na literatura. / Modeling and numerical simulation of solutes (e.g. Tracers) in heterogeneous and anisotropic porous media such as aquifers and oil reservoirs, constitute a bigger challenge of mathematics and numerical nature. Modeling sealants faults, channels, inclined wells, pinch outs and other complex features of these geological formations demand the use of unstructured and not orthogonal meshes, able to adapt naturally to the domain under study. The computational packages used commonly in the oil industry, mostly, are based on the Finite Difference Method with Two Point Flow Approximation (TPFA) and the Amount First Order Upwind method (FOU), due to its ease of implementation and its computational efficiency. Unfortunately, TPFA methods are unable to produce conver-gent solutions in non-orthogonal meshes or in permeability or dispersion full Tensor and FOU methods produce solutions with excessive numerical diffusion, requiring excessively refined mesh to obtain reliable solutions. An interesting alternative to TPFA, which allows the use of full tensor and not orthogonal meshes, is the Galerkin Finite Element Method (FEM), but this method does not produce solutions locally conservative, which can be a serious problem for modeling problems involving conservation laws as the flow in porous media. An interesting alternative is the Finite Volume Methods (MVF). In its most robust embodiments, these methods are able to cope with any polygonal mesh and full permeability or dispersion tensors and with an arbitrary anisotropy ratio, beyond producing discrete approximations of high order and locally conservative. In this context, the present study, we present one MVF formulation cell centered to modeling the transport of a non-reactive tracer in single-phase flow in heterogeneous and anisotropic porous media. For the elliptical discretization terms, both, the pressure equation as the equation advection-dispersion-reaction (ADRE), we used The FVMF multipoint flow approximation that uses the diamond stencil (MPPA-D) and for the discretization of hyperbolic terms, we use the FOU method and an MVF type MUSCL (Monotone Upstream Centered Scheme for Conservation Laws). In order to test our formulation, we solve some benchmark problems in the literature.

O bulbo molhado formado na irrigação por gotejamento pode ser modelado a partir informações das propriedades físico-hídricas do solo. Mas apesar do grande progresso na modelagem, a aplicação de modelos na irrigação e drenagem ainda não foi implementada em nenhum nível de maneira substancial. Assim, o objetivo do presente trabalho foi desenvolver um modelo numérico utilizando a técnica dos volumes finitos para estimar a forma e as dimensões do volume de solo molhado sob irrigação por gotejamento, e verificar a validade do modelo através da comparação com dados recolhidos em condições experimentais. O ensaio foi conduzido na área experimental do Departamento de Engenharia de Biossistemas da Escola Superior de Agricultura Luiz de Queiroz-ESALQ/USP. As sondas de TDR foram confeccionadas segundo procedimentos descritos por Souza et al. (2006). O perfil do bulbo molhado formado no solo abaixo de um emissor do tipo gotejador foi determinado para 3 tipos de material: areia, solo arenoso e solo argiloso e para 3 vazões nominais do emissor: 2, 4 e 6 L/h. Cada uma das combinações entre tipo de solo e vazão foi repetida três vezes em caixas (1000 L) diferentes, totalizando 27 medições. Um modelo numérico foi desenvolvido para simular a distribuição da água no perfil do solo abaixo de uma fonte pontual. A solução da equação diferencial para o movimento da água em solo não saturado foi realizada pela discretização do espaço-tempo utilizando a técnica dos volumes finitos. Um algoritmo em linguagem Visual Basic foi escrito para implementar o conjunto de equações e simular a evolução do bulbo molhado no tempo. Foi simulada a formação do bulbo molhado nas mesmas condições do experimento, e gráficos de isolinhas de umidade foram traçados no software Surfer. O modelo numérico proposto foi capaz de simular a formação do bulbo molhado em diferentes condições de tipo de solo e vazão do emissor. / The soil water patterns in drip irrigation can be modeled from physical and hydraulic properties of soil. But despite the great progress in modeling, the models have not been implemented at any level in irrigation and drainage. The objective of this study was to develop a numerical model using finite volume technique to estimate the shape and dimensions of the wetted soil volume under drip irrigation and verify the validity of the model by comparing data collected under experimental conditions. The trial was conducted at the experimental site of the Department of Biosystems Engineering, Escola Superior de Agricultura Luiz de Queiroz- ESALQ/USP. The TDR probes were manufactured according Souza et al. (2006). The soil water patterns under a drip emitter type was determined for three types of material: sand, sandy soil and clay soil and three nominal flows of the emitters: 2, 4 and 6 L/h. Each of the combinations of soil type and flow rate was repeated three times in boxes (1000 L), totaling 27 measurements. A numerical model was developed to simulate the distribution of water in the soil profile below a point source. The solution of the differential equation for the movement of water in unsaturated soil was carried out by the discretization of space-time using the technique of finite volume. An algorithm in Visual Basic language was written to implement the set of equations and simulate the evolution of wetted soil volume in time. Contour plots of soil water content were drawn in Surfer software. The proposed numerical model was able to simulate wetted soil volume under different conditions of soil type and flow of the emitter.