Modéliser les océans du globe

National audienceIl existe, sous la surface de nos océans, un immense réseau de courants marins, véritables tapis roulants des mers, qui transportent des masses d'eau absolument gigantesques. Ces courants à grande échelle, parmi lesquels figure le célèbre Gulf Stream, jouent un rôle primordial dans la dynamique des océans, et bien entendu dans l'équilibre thermo-dynamique de notre planète. Il est donc essentiel de pouvoir les comprendre afin d'anticiper d'éventuels déséquilibres qui pourraient advenir, par exemple, dans le cadre du réchauffement climatique

Analysis of a projection method for low-order non-conforming finite elements

International audienceWe present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions

On a projection method for piecewise-constant pressure nonconforming finite elements

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree nonconforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions

A DDFV method for a Cahn-Hilliard/Stokes phase field model with dynamic boundary conditions

International audienceIn this paper we propose a "Discrete Duality Finite Volume" method (DDFV for short) for the diffuse interface modelling of incompressible flows. This numerical method is, conservative, robust and is able to handle general geometries and meshes. The model we study couples the Cahn-Hilliard equation and the unsteady Stokes equation and is endowed with particular nonlinear boundary conditions called dynamic boundary conditions. To implement the scheme for this model we have to define new discrete consistent DDFV operators that allows an energy stable coupling between both discrete equations. We are thus able to obtain the existence of a family of solutions satisfying a suitable energy inequality, even in the case where a first order time-splitting method between the two subsystems is used. We illustrate various properties of such a model with some numerical results

An adaptive pressure correction method without spurious velocities for diffuse-interface models of incompressible flows

International audienceIn this article, we propose to study two issues associated with the use of the incremental projection method for solving the incompressible Navier-Stokes equation. The first one is the combination of this time splitting algorithm with an adaptive local refinement method. The second one is the reduction of spurious velocities due to the right-hand side of the momentum balance. We propose a new variant of the incremental projection method for solving the Navier-Stokes equations with variable density and illustrate its behaviour with the example of two phase flows simulations using a Cahn-Hilliard/Navier-Stokes model

An explicit MUSCL scheme on staggered grids with kinetic-like fluxes for the barotropic and full Euler system

International audienceWe present a second order scheme for the barotropic and full Euler equations. The scheme works on staggered grids, with numerical unknowns stored at dual locations, while the numerical fluxes are derived in the spirit of kinetic schemes. We identify stability conditions ensuring the positivity of the discrete density and energy. We illustrate the ability of the scheme to capture the structure of complex flows with 1d and 2d simulations on mac grids

An asymptotic preserving scheme on staggered grids for thebarotropic Euler system in low Mach regimes

International audienceWe present a new scheme for the simulation of the barotropic Euler equation in low Mach regimes. The method uses two main ingredients. First, the system is treated with a suitable time splitting strategy, directly inspired from [J. Haack, S. Jin, J.-G. Liu, Comm. Comput. Phys., 12 (2012) 955-980], that separates low and fast waves. Second, we adapt a numerical scheme where the discrete densities and velocities are stored on staggered grids, in the spirit of MAC methods, and with numerical fluxes derived form the kinetic approach. We bring out the main properties of the scheme in terms of consistency, stability, and asymptotic behaviour, and we present a series of numerical experiments to validate the method

An explicit MUSCL scheme on staggered grids with kinetic-like fluxes for the barotropic and full Euler system

International audienceWe present a second order scheme for the barotropic and full Euler equations. The scheme works on staggered grids, with numerical unknowns stored at dual locations, while the numerical fluxes are derived in the spirit of kinetic schemes. We identify stability conditions ensuring the positivity of the discrete density and energy. We illustrate the ability of the scheme to capture the structure of complex flows with 1d and 2d simulations on mac grids

Modeling and Computation of a liquid-vapor bubble formation

International audienceThe Capillary Equation correctly predicts the curvature evolution and the length of a quasi-static vapour formation. It describes a two-phase interface as a smooth curve resulting from a balance of curvatures that are influenced by surface tension and hydrostatic pressures. The present work provides insight into the application of the Capillary Equation to the prediction of single nu-cleate site phase change phenomena. In an effort to progress towards an application of the Capillary Equation to boiling events, a procedure to generating a numerical solution, in which the computational expense is reduced, is reported

Kinetic schemes on staggered grids for barotropic Euler models: entropy-stability analysis

International audienceWe introduce, in the one-dimensional framework, a new scheme of finite volume type for barotropic Euler equations. The numerical unknowns, namely densities and velocities, are defined on staggered grids. The numerical fluxes are defined by using the framework of kinetic schemes. We can consider general (convex) pressure laws. We justify that the density remains non negative and the total physical entropy does not increase, under suitable stability conditions. Performances of the scheme are illustrated through a set of numerical experiments