Advances in the development of the Fokker-Planck method for simulation of rarefied gases

Abstract

Flows encountered in space applications, like atmospheric reentry or jet plume interactions, are characterized by a wide range of the Knudsen number. The well-known Boltzmann equation describes the evolution of a distribution function f in such rarefied flows due to intermolecular collisions: A common approach to numerically solve Eq. (1) is the Direct Simulation Monte-Carlo (DSMC) method pioneered by Bird [1]. The method is very efficient for high Knudsen numbers but becomes computationally intensive when approaching the continuum limit. Approximations of the collision operator S Boltz in Eq. (1) can greatly reduce this computational cost. One such approximation is the Fokker-Planck collision operator S FP [2]: where the drift coefficient A i and the diffusion coefficient D of Eq. (2) are model parameters chosen in such a way that moments calculated using the Boltzmann collision operator are reproduced in the continuum limit. The resulting Fokker-Planck equation can be solved through stochastic motion which in turn can be modelled using a particle method. Due to the similarity in their formulation a hybrid method based on DSMC and FP can be derived which allows the computationally efficient simulation of flows with a broad range of Knudsen numbers [3]. For the application of the FP model to engineering problems the simulation of complex gases and gas mixtures must be possible. The FP model has been extended from monatomic to diatomic gas for single species [4, 5] as well as for mixtures [6, 7]. However, the extension of the model to polyatomic gases for single and multispecies applications is still a research topic. This paper discusses how the recently developed FP models for diatomic gases can be extended to allow modeling of polyatomic molecules using the Master-equation ansatz. We further point out why the current multi-species formulations [6, 7] do not contain the special case of a single-species gas [8] and propose modifications to establish consistency.