28th ICPIG, July 15-20, 2007, Prague, Czech Republic

Numerical simulation of filamentary discharges with the parallel adaptive mesh refinement technique

S. Pancheshnyi1 , P. Ségur1 and A. Bourdon2

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1P Laboratoire Plasma et Conversion d'Energie, CNRS UMR 5213

Université Paul Sabatier, 118, route de Narbonne 31062 Toulouse, France

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2P Laboratoire Énergétique Moléculaire et Macroscopique, Combustion, CNRS UPR 288 École Centrale Paris,

Grande Voie des Vignes 92295 Châtenay-Malabry, France

Direct simulation of filamentary gas discharges like streamers or DBD microdischarges needs to use an adaptive mesh. The objective of this paper is to develop a strategy which can use a set of grids with suitable local refinements for the continuity equations and the Poisson equation in 2D and 3D geometry with a high-order discretization. The advantages of the approach are presented with a filamentary discharge simulation in a plane-plane geometry in nitrogen within the diffusion-drift approximation.

1. Introduction

Simulation of high-pressure gas discharges is important for numerous plasma applications. Present work is a part of a project directed towards the development of a general numerical code capable of simulating discharge dynamics and discharge coupling with reactive gas flows, taking into account the solution of coupled multidimensional flow equations, electric field equation, and nonequilibrium kinetics.

2. The model

The motion of charged particles is described by the balance equation (1) within the drift-diffusion approach (2)

∂nk

∂tdiv jk=Sk

jk= vknk−Dk∇ nk

Here, nk and qk represents density and charge of each kind of charged species (electrons and different sorts of ions); flux jk consists of advective part with drift velocity vk and diffusion part with coefficient Dk. The right part in the balance equation Sk corresponds to various plasma-chemical processes like ionization, recombination, electron attachment, and photoionization, etc.

The system of equations (1,2) is coupled with an equation for electric field E distribution which could be written in differential form as

∇ E=∑kqknk /0

Here, 0 is the electric permittivity of free space. Equation (3) could be solved through potential distribution (r) by the Poisson's equation.

The discretization of the advective term in the balance equation (1) requires care. Numerical results [1] show that a 3rd order upwind scheme QUICKEST [2], combined with the ULTIMATE flux limiter [3] is competent for a filamentary discharge simulation avoiding valuable numerical oscillations and diffusion. At the same time, a 2nd order central scheme is used for species diffusion computation.

A 4th order compact solver for the Poisson's equation (3) has been developed by using the multi-grid and conjugate-gradient techniques.

The equations were solved in processes- and time- splitting fashion and a two-step “predictor-corrector” technique was used to achieve a 2nd order accuracy in time.

2. Adaptive mesh refinement

The Adaptive Mesh Refinement (AMR) is a method of adaptive meshing in which it discretizes the continuous domain of interest into a grid of many individual elements. It was already applied successfully for streamer simulation [4,5].

In the AMR we start with a base coarse grid. As the solution proceeds we identify the regions requiring more resolution by some parameter characterizing the solution, the local truncation error. We superimpose finer subgrids only on these regions. Finer and finer subgrids are added recursively until either a given maximum level of refinement is reached or the local truncation error has dropped below the desired level. Thus in an adaptive mesh

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

refinement computation grid spacing is fixed for the base grid only and is determined locally for the subgrids according to the requirements of the problem.

The PARAMESH software used in this work was developed at NASA Goddard Space Flight Center under the HPCC and ESTO/CT projects [6]. It is a package of Fortran 90 subroutines designed to provide an application developer with an easy route to extend an existing serial code which uses a logically cartesian structured mesh into a parallel code with AMR.

In our implementation, the decision whether a finer or a coarse grid should be used on a certain region is made with respect to ni/ni+1 electron density and electric field Ei/Ei+1 jump on the interface of each couple of cells i/i+1

Cnrefni /n i1Cn

deref

CErefE i /E i1CE

deref

In the next simulations the above mentioned values were fixed to Cn

ref = 1.5, Cnderef = 3.0, CE

ref = 1.05 and CE

deref = 1.15 for every cell with non-zero values.

Conservative 1st and 2nd order interpolations were used here for data restriction and prolongation, respectively. Interpolation occurs during the prolongation or restriction operations which fill either newly created blocks with data in case of prolongation, and when filling guards cells at block boundaries next to less refined neighbor blocks.

3. Simulation of a streamer development in nitrogen

An example of streamer propagation in pure nitrogen in a 1-cm plane-plane gap in a uniform external field of 150 Td is presented in this Section. 2D cylindrical symmetric and 3D cartesian formulations were used in the simulation.

The photoionization has not been treated in this case while the photoemission (ph = 510-3) and the ion-electron emission (ei = 10-1) have been taken into account as a source of secondary electrons. The photoemission produced by 2+ system of nitrogen was computing using direct technique.

The discharge was initiated by a quasi-neutral preionization gaussian spot with n0 = 3.5 1010 cm-3

peak density and = 10-2 cm spatial dispersion that gives about 105 charges in total in the spot.

The kinetic model consists of 35 reactions and includes 11 components: neutral particles N2, N, N2(A), N2(B), N2(a'), N2(C); ions N+, N2

+, N3+, N4

+; and electrons. The rate constants of reactions including electrons are taken as functions of the reduced field [7]. The rate constants of ion-molecule reactions and processes that include radicals and excited molecules are taken from [8].

3. The discharge dynamics

Initially, an ordinary avalanche slowly propagates from the spot toward the positive electrode. At a same distance, named the avalanche-to-streamer transition length, a negative streamer forms as it is shown in Figure 1.

The avalanche-to-streamer transition occurs in the case in the middle of the gap, i.e. at xcr 0.5 cm that is in a good agreement with the Meek-Raether criterion xcr

M-R = 0.47 cm [9]

xcrM−R

≈20−lnN0

where and N0 are the first ionization coefficient and the total number of initial electrons, respectively.

As soon as the first streamer propagating from the negative electrode touches the opposite electrode, a reflected ionization wave - positive streamer - starts to propagate toward the negative electrode after a delay. This step as well as the following high-conductivity plasma channel formation is presented in Figure 1.

We note that the mesh structure was analyzed and modified if it was necessary each 5 time steps and schematically illustrated in Figures 1. The computational blocks of 8 8 uniform cells are presented in the Figures.

This adaptive mesh allows to use an effective resolution up to 10-3 cm in the streamer head region with only 300-800 blocks (i.e. (2-5)104 cells) that corresponds to 106 cells on a uniform grid. This advantage is even more pronounced in 3D case.

The discharge development corresponding to the same conditions was simulated in 3D cartesian geometry as well. In this case the peak number of blocks needed for simulations increases up to about 9103 (i.e. up to 5106 cells) that corresponds to 109

cells on a uniform grid. Note, the total number of the cells in 2D and 3D cases differs by a factor of 100 that leads to much longer computational time.

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

Electron density distribution is presented in Figure 2 at time moments 30 and 75 ns. Every block presented in the Figure consists of 8 8 8 uniform cells.

In spite of the identical numerical technique used in 2D and 3D simulations here, the results are not exactly the same for both cases. The peak value of the electric field which has the biggest difference is presented in Figure 3. Both traces in the Figure have similar shape; all stages discussed above can be easily detected there. Nevertheless, a time and amplitude shift is present.

Figure 1:  Discharge dynamics  at  different   times.  From  top to bottom at 10, 50, 75 and 78 ns. Iso­lines of electron  density (in cm­3), reduced electric field (in Td), net charge  (in cm­3)  and normilized potential  are presented  in the  columns.

Figure   2:   Negative   streamer   formation   and   backward  positive   streamer   propagation   at   30   ns   and   75   ns,  respectively. Iso­surfaces of electron density (in cm­3) and  block bounds are presented.

28th ICPIG, July 15-20, 2007, Prague, Czech Republic

The distinction for other computed parameters (e.g., speed of propagation, external current presented in Figure 4, electron density, etc.) is much less pronounced.

4. Conclusions

A numerical code which uses an adaptive grid refinement strategy for the computation of filamentary discharges within the diffusion-drift approximation in 2D and 3D geometry was developed. The numerical discretizations are based on finite volume methods and have 2nd order accuracy in space and time.

As an example, a filamentary discharge was simulated in plane-plane geometry in nitrogen. The main stages of the discharge development are presented up to the moment of spark formation.

References

1] A. Bourdon, D. Bessieres, J. Paillol, A. Michau, K. Hassouni, E. Marode, and P. Ségur. Influence of numerical schemes on positive streamer propagation. In Proceedings of the 27nd Int. Conf. on Phenomena in Ionized Gases, pages 17-422, Eindhoven, The Netherlands, 2005.

[2] B. P. Leonard. A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Computer methods in applied mechanics and engineering, 19:59, 1979.

[3] B. P. Leonard. The ULTIMATE conservative difference scheme applied to unsteady one- dimensional advection. Computer methods in applied mechanics and engineering, 88:17, 1991.

[4] C. Montijn, W. Hundsdorfer, and U. Ebert. An adaptive grid renement strategy for thesimulation of negative streamers. Journal of Comput. Phys., 219:801, 2006.

[5] D. Nikandrovand, L. Tsendin, R. Arslanbekovand, and V. Kolobov. Dynamics of ionization fronts during high-pressure gas breakdown. In Proceedings of the 59th Annual Gaseous Electronics Conference, volume SRP2, page 31, Columbus, Ohio, 2006.

[6] P. MacNeice, K. M. Olson, C. Mobarry, R. Fainchtein, and C. Packer. PARAMESH : A parallel adaptive mesh refinement community toolkit. Computer Physics Communications, 126:330, 2000.

[7] Yannick Cesses. Two-dimensional model of a DBD: study of electrodes phenomena. PhD thesis, CPAT, Université Paul Sabatier, Toulouse, France, 2004.

[8] M. Capitelli, C. M. Ferreira, B. F. Gordiets, and A. I. Osipov. Plasma Kinetics in Atmospheric Gases. Springer, Berlin, 2000.

[9] Y. P. Raizer. Gas Discharge Physics. Springer- Verlag, Berlin, 1991.

Figure 3: Peak value of the electric field Figure 4: Electric current profile