Modélisation du transport de soluté en milieux poreux par la méthode d’éléments finis mixtes hybrides – développement d’un limiteur de flux

Une méthode d’éléments finis mixte hybride est appliquée pour l’approximation de l’écoulement associé au transport en milieu poreux non saturé. Le développement de ce modèle s’effectue dans le cadre du projet européen ARWET, lequel a pour objectif l’étude de nouvelles potentialités de dissipation des pesticides dans les zones humides. La formulation du modèle bidimensionnel est fondée sur les propriétés de l’espace de Raviart-Thomas. L’écueil numérique que posent les problèmes à convection dominante est surmonté par l’introduction d’un limiteur de flux alors qu’un limiteur de pente est généralement utilisé dans la littérature. Le limiteur suggéré est inconditionnellement stable et permet de conserver la précision des résultats à nombre de Peclet élevé.A mixed hybrid finite element method was applied to obtain a numerical approximation of the flow and associated transport equations in unsaturated media. The model was developed under the framework of the European Life Environment project ARTWET, which aims to study new treatment potentials for the mitigation of non-point source pesticide pollution in a constructed wetland. The model formulation used is based on Raviart-Thomas space properties, considering a two-dimensional domain divided into triangular elements. In order to control for the difficulties when convection is the dominant process, a flux limiting tool was introduced, although a slope limiter is generally used in the literature. The suggested flux limiting tool makes it possible to preserve precision and unconditional stability at low and very high Peclet numbers

Numerical Reliability and CPU Time for the Mixed Methods applied to Flow Problems in Porous Media

This work is devoted to the numerical reliability and time requirements of the Mixed Finite Element (MFE) and Mixed-Hybrid Finite Element (MHFE) methods. The behavior of these methods is investigated under the influence of two factors: the mesh discretization and the medium heterogeneity. We show that, unlike the MFE, the MHFE "suffers" with the presence of flatted triangular elements. A numerical reliability analyzing software (Aquarels) is used to detect the instability of the matrix-inversion code generated by MAPLE which is used in the MHFE code. We also show that the spectral condition number of the algebraic systems furnished by both methods in heterogeneous media grows up linearly according to the smoothness of the hydraulic conductivity. Furthermore, it is found that the MHFE could accumulate numerical errors if the conductivity varies abruptly in space. Finally, we compare running-times for both algorithms by giving various numerical experiments

On the Finite Volume Reformulation of the Mixed Finite Elements Method on Triangles

We analyse the finite volume reformulation of the triangular mixed finite element approximation for the porous flow equation, as proposed in [10] [9]. We show that the finite volumes are obtained by aggregation of finite elements (usually one, sometimes two or more), that the matrix of the finite volume equations is regular, but generally not symmetrical, and that the finite volume formulation is algebraically equivalent to the mixed approximation. The finite volume matrix becomes symmetrical in the stationary case, and positive definite when the triangulation satisfies the Delaunay condition

New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes

In this paper we introduce an extension of Van Leer's slope limiter for two-dimensional Discontinuous Galerkin (DG) methods on arbitrary unstructured quadrangular or triangular grids. The aim is to construct a non-oscillatory shock capturing DG method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion. Unlike some splitting techniques that are limited to linear approximations on rectangular grids, in this work, the solution is approximated by means of piecewise quadratic functions. The main idea of this new reconstructing and limiting technique follows a well-known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution. Numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of the proposed slope limiters

Multi-Step Level-Set Ice Accretion Simulation with the NSMB solver

Icing effects can reduce the flight safety under certain weather conditions. According to the US National Transport Safety Board, icing is one of the major causes of flight accidents. Supercooled water droplets present in clouds impinge on the surface of aircraft structures. They either solidify totally on impact or partially then creating a thin liquid film runback depending on the flow temperature and speed hence, creating dry rime ice or glaze wet ice respectively. Designing an adequate de-icing mechanisms requires full knowledge of the icing phenomenon itself. Icing experimental study cannot exceed the scope of a handful of simple cases due to complexity and cost. Consequently the use of computational fluid dynamics is justified. The icing process is assumed broken up into four steps: 1) single phase air flows around the wing 2) transporting water suspended droplets; droplets impinge into the surface 3) generating a liquid or dry film exchanging energy with the surface 4) accreated to shape the final form during a certain exposure time. This process is usually assumed to occur on a single step considering that the time scale of the icing process is very long compared with that of the air flow. Current Icing simulation codes used by industries are based on over-simplified models. 1) A 2D inviscid panel methods with an empirical boundary layer method is used for the air flow. Which is usually followed by 2) a Lagrangian transport of droplets. And finally 3,4) an iterative thermodynamic model for the liquid film to compute the ice thickness. To generate the final geometry however, a Lagrangian node displacement is needed. A multi-step icing approach repeats this process for portions of the required exposure time but still with decoupled time scales. Maintaining a good grid quality requires a tedious amount of work, since strange irregularities in iced shapes are difficult to be fully accounted for. The Level-Set method introduced by Osher and Fedkiw could alleviate such a task. A passive scalar function is introduced and is put equal to zero at the interface, positively defined outside and negatively inside; the zero level represents the time evolution of the air/ice interface. To complete the model, a PDE type thermodynamic model is used for the film, coupled with an external flow solver. In the present study a new method of icing simulation is developed. To get the most out of such model, it is developed in the three-dimensional structured multiblock Navier-Stokes solver NSMB. For a multi-step icing procedure, the geometry is defined by a passive scalar called the level-set. This level-set function is set equal to the distance, negative on the inside and positive outside. A penalized Navier-Stokes equation is solved on the external flow using a simple non-body fitted mesh, wherein the solid is represented by the negative level-set valued cells. The droplets are transported using an Eulerian approach using a TVD and a local time stepping schemes. The impingement rate or what's called the collection efficiency is then fed to a Shallow-Water Icing Model that evaluates the ice accretion, its height and velocity. The convective heat transfer coefficient is obtained from the Navier-Stokes solver. Following that the Level-set function is advected with the icing velocity to predict the new deformed geometry. The process is then repeated for as many portions of the exposure time as needed

Assessement of velocity fields through open-channel flows with an empiric law

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