Stéphane Brull

Local discrete velocity grids for deterministic rarefied flow simulations

Most of numerical methods for deterministic simulations of rarefied gas flows use the discrete velocity (or discrete ordinate) approximation. In this approach, the kinetic equation is approximated with a global velocity grid. The grid must be large and fine enough to capture all the distribution functions, which is very expensive for high speed flows (like in hypersonic aerodynamics). In this article, we propose to use instead different velocity grids that are local in time and space: these grids dynamically adapt to the width of the distribution functions. The advantages and drawbacks of the method are illustrated in several 1D test cases.

Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model

The present work is devoted to the simulation of a strongly magnetized plasma considered as a mixture of an ion fluid and an electron fluid. For the sake of simplicity, we assume that the model is isothermal and described by Euler equations coupled with a term representing the Lorentz force. Moreover we assume that both Euler systems are coupled through a quasi-neutrality constraint of the form

Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation

n_{i}=n_{e}

A numerical adaptative method for solving kinetic equations based on local velocity grids

. The numerical method which is described in the present document is based on an Asymptotic-Preserving semi-discretization in time of a variant of this two-fluid Euler-Lorentz model with a small perturbation of the quasi-neutrality constraint. Firstly, we present the two-fluid model and the motivations for introducing a small perturbation into the quasi-neutrality equation, then we describe the time semi-discretization of the perturbed model and a fully-discrete finite volume scheme based on it. Finally, we present some numerical results which have been obtained with this method.

Boundary conditions for the Boltzmann equation for rough walls

It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a Discrete Velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro and nano-flows.

Two dimensional local adaptive discrete velocity grids for rarefied flow simulations

The aim of this paper is to present a new numerical deterministic method for solving kinetic equations using local dynamical grids for the velocity variable. Indeed, classical methods are based on a global velocity grid constructed according to initial conditions. But these classical methods present a huge computational cost and realistic cases are difficult to simulate. In the present method, the dynamical grids are constructed by solving firstly a system of conservation laws and next by considering a projection of the distribution function on the new grid. The projection is performed by using interpolation procedures involving essentially non oscillatory techniques to diminish oscillations. During the presentation, we will show that this method is in particular well adapted to physical situations with sharp gradients of temperatures.

Problem of evaporation-condensation for a two component gas in the slab

In some applications, rarefied gases have to considered in a domain whose boundary presents some nanoscale roughness. That is why, we have considered (Brull,2014) a new derivation of boundary conditions for the Boltzmann equation, where the wall present some nanoscale roughness. In this paper, the interaction between the gas and the wall is represented by a kinetic equation defined in a surface layer at the scale of the nanometer close to the wall. The boundary conditions are obtained from a formal asymptotic expansion and are describded by a scattering kernel satisfying classical properties (non-negativeness, normalization, reciprocity). Finally, we present some numerical simulations of scattering diagrams showing the importance of the consideration of roughness for small scales in the model.

Gas-surface interaction and boundary conditions for the Boltzmann equation

We propose a deterministic method designed for unsteady flows, based on a discretization of the Boltzmann (BGK) equation with local adaptive velocity grids. These grids dynamically adapt in time and space to the variations of the width of the distribution functions. This allows a significant reduction of the memory storage and CPU time, as compared to standard discrete velocity methods, and avoid the delicate problem to construct a priori a suffcient global velocity grid.We propose a deterministic method designed for unsteady flows, based on a discretization of the Boltzmann (BGK) equation with local adaptive velocity grids. These grids dynamically adapt in time and space to the variations of the width of the distribution functions. This allows a significant reduction of the memory storage and CPU time, as compared to standard discrete velocity methods, and avoid the delicate problem to construct a priori a suffcient global velocity grid.