A Fokker-Planck model of hard sphere gases based on H-theorem

Abstract

It has been shown recently that the Fokker-Planck kinetic model can be employed as an approximation of the Boltzmann equation for rarefied gas flow simulations [4, 5, 10]. Similar to the direct simulation Monte-Carlo (DSMC), the Fokker-Planck solution algorithm is based on the particle Monte-Carlo representation of the distribution function. Yet opposed to DSMC, here the particles evolve along independent stochastic paths where no collisions need to be resolved. This leads to significant computational advantages over DSMC, considering small Knudsen numbers [10]. The original Fokker-Planck model (FP) for rarefied gas flow simulations was devised according to the Maxwell type pseudo-molecules [4, 5]. In this paper a consistent Fokker-Planck equation is derived based on the Boltzmann collision integrals and maximum entropy distribution. Therefore the resulting model fulfills the H-theorem and leads to correct relaxation of velocity moments up to heat fluxes consistent with hard sphere interactions. For assessment of the model, simulations are performed for Mach 5 flow around a vertical plate using both Fokker-Planck and DSMC simulations. Compared to the original FP model, significant improvements are achieved at high Mach flows.