Browsing by Author "D'Andrea, Danilo"

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Modelling of intra- and inter species charged particle collisions for flow simulation in pulsed plasma thrusters
(2008) D'Andrea, Danilo; Munz, Claus-Dieter (Prof. Dr.)
A better physical understanding of electrical space propulsion systems like Pulsed Plasma Thrusters requires the numerical modelling and simulation of highly rarefied plasma flows. Mathematically, such phenomena demand a kinetic description which is established by the complete, time-dependent Boltzmann equation. An attractive numerical approach to tackle this complex non-linear problem consists of a combination of the well-known Particle-in-Cell (PIC) and Monte Carlo methods extended by a PIC-based Fokker-Planck solver, on which we focus our attention in the following. This numerical model accommodates the physics of interaction of charged particles with electromagnetic fields, inelastic electron-neutral scattering as well as intra- and inter-species charged particle Coulomb collisions. To describe elastic intra- and inter-species charged particle Coulomb collisions it is convenient to start from the Boltzmann collision integral with the classical Rutherford differential cross section. A Taylor series expansion up to second order in velocity of the post-collision distribution functions and the adoption of a cut-off value for the impact parameter permits the final integration of the Boltzmann integral to obtain the Fokker-Planck equation. The central quantities appearing in the Fokker-Planck equation are the friction force vector and the diffusion tensor. The keys to compute the friction and diffusion coefficients are the Rosenbluth potentials which are in turn complicate integrals of the field particle distribution function and the relative velocity between test and field particles. Usually, strong assumptions like isotropic velocity distribution of the scatterer, are made to evaluate the Rosenbluth potentials. Observing that the Rosenbluth potentials are convolution intergrals addresses the use of fast Fourier transform techniques to calculate these quantities and their derivatives rapidly with the advantage of being free of any additional assumption. Furthermore, such a determination the Rosenbluth potentials is the basis to model collisional relaxation in a complete self-consistent manner. In order to fit the three-dimensional Fokker-Planck equation of the scattered distribution function into a particle-based method framework, the equivalence with the stochastic differential equation (SDE) is exploit. The stochastic variable C(t) which obeys the SDE is later identified with the charged particle velocity. Also in this context the friction force vector and a matrix derived from the diffusion tensor play the central role. By means of Ito-Taylor expansion and Ito calculus the stochastic differential equation is discretised and numerical schemes are derived. In this work, explicit weak schemes up to approximation order two have been applied to update the particles velocity. These weak Ito-Taylor schemes together with the Fourier transform method and particle-mesh interface techniques form a remarkable simulation tool to study collisional relaxation processes from first principles. For instance by means of this tool, a more realistic evaluation of the time scales can be provided since the classical test-particle approach is not necessary anymore thanks to self-consistency. The introduced intra-species charged particle modelling can be easily adapted for inter-species electron-ion particle collisions. Finally, the structure of the developed PIC-based method to solve the Fokker-Planck equation also allows to combine intra- and inter-species collisions to perform coupled simulations.