Partenaires

CNRS
Logo INP/ EINSEEIHT
Logo UPS


    Version Française         English Version       

Rechercher

Sur ce site

Accueil du site > Groupes de recherche > Groupe de Recherche Energétique, Plasmas et Hors Equilibre > Projets en cours > Méthodes asynchrones pour la simulation de décharges à pression atmosphérique

Asynchronous methods for the simulation of gas discharges at atmospheric pressure

25 février 2008

GREPHE Members Research Topics Ongoing Projects News Publications

Participants

Students : T. Unfer

Permanents : J.P. Boeuf

Partners, collaboration and contracts

Collaboration with ONERA/DMAE, ONERA/DTIM

T. Unfer benefits from a fellowship from the EADS Foundation.

Introduction

The description of an atmospheric plasma is a multi-scale problem in both time and space which makes it computationally hard to solve. Hence the timescales at stake can range from the picosecond for electron motion and dielectric relaxation up to the millisecond for the aerodynamic flow control application. Furthermore the dimension of the plasma sheath tends to shrink drastically due to the high density of charged particles which implies large restriction on the size of the grid spacing. At atmospheric pressure the plasma tends to be filamentary which means that the fast evolving phenomena are very localized whereas large areas of the simulated domain have quite low activity. An innovative numerical approach is being developed to take advantage of this property. The goal is to concentrate the computation effort on the fast evolving zones without having to spend much CPU on the slow areas. The idea is to refresh independently each interface flux or source with respect to a global clock rather than recomputing every fluxes and sources from one time step to the other. The method is therefore called "asynchronous" because computation of the fluxes are not synchronized.

Asynchronous numerical integration method for transport equations

Explicit integration techniques face a severe time step restriction known as the Courant-Friedrich-Levy (CFL) condition. For the system to remain numerically stable this condition has to be imposed over the whole mesh, using classical methods. As a result the integration time step of the system is limited by the minimum of the CFL conditions all over the domain.

Let us consider a conservation law in 1D for simplicity

\frac{\partial n}{\partial t}+\frac{\partial\Gamma}{\partial x}=S

\Gamma=nv(x) with v(x)\geq0 a given advection velocity and S some source term In this case the first-order upwind scheme over a regular mesh would be :

\left\{ \begin{array}{ll}n_{i}^{k+1}=n_{i}^{k}-\frac{\Delta t}{\Delta x}(n_{i}^{k}v_{i+\frac{1}{2}}-n_{i-1}^{k}v_{i-\frac{1}{2}})+S_{i}\Delta t\\ t_{k+1}=t_{k}+\Delta t\end{array}

n_{i} are the density values at the cell centers x_{i}=i\Delta x , v_{i+\frac{1}{2}} is the velocity values taken at the cell interfaces x_{i+\frac{1}{2}}=(i+\frac{1}{2})\Delta x . For the standard scheme \Delta t is the minimum of the CFL condition all over the mesh.

The asynchronous scheme assumes that each flux or source term is updated independently according to a refresh time tag with respect to a global simulation clock. Over some user defined time step the solution is built as follows :

\left\{\begin{array}{ll}n_{i}^{k+1}=n_{i}^{k}-\frac{1}{\Delta x}(\sum_{p}\Delta t_{\Gamma+}^{p}\Gamma_{i+\frac{1}{2}}(t_{p})-\sum_{q}\Delta t_{\Gamma-}^{q}\Gamma_{i-\frac{1}{2}}(t_{q}))+\sum_{r}\Delta t_{S}^{r}S_{i}(t_{r})\\ t_{k+1}=t_{k}+\Delta t_\end{array}

For the scheme to be stable each local time step is limited by the local value of the CFL condition. It has also been shown that the asynchronous scheme has nice properties in term of numerical diffusion.

Application to surface Dielectric Barrier Discharge (DBD) at atmospheric pressure

DBB geometry

The above Figure shows a typical DBD configuration that has been suggested for aerodynamic flow control. In a surface DBD plasma actuator, a sinusoidal voltage is applied between the electrodes. Transient discharges develop above the dielectric surface and momentum transfer between charged particles and neutral molecules can generate a flow or modify the boundary layer of a flow along an airfoil. A numerical comparison between the asynchronous scheme and standard explicit scheme is shown below : response to a voltage step in pure Nitrogen. Both methods give very close results and CPU time is reduced by more than ten using the asynchronous method.

Electron density comparison for asynchronous scheme (left) and standard explicit scheme (right)
Ion density comparison for asynchronous scheme (left) and standard explicit scheme (right)
Discharge current comparison for asynchronous scheme (blue) and standard explicit scheme (red)
CPU time ratio between standard and asynchronous methods

Publications

- T. Unfer, J.P. Boeuf, F. Rogier, F. Thivet, "An asynchronous scheme with local time stepping for multi-scale transport problems : Application to gas discharges", J. Comp. Phys. 227, 2007, 898