A stochastic particle Fokker–Planck method with nonlinear production terms for a variable hard-sphere gas

Abstract

The stochastic particle Fokker–Planck (FP) method has been gaining increasing attention in the field of rarefied gas dynamics due to its potential to reduce the computational costs of the direct simulation Monte Carlo method. The FP method approximates the discrete binary collisions of the Boltzmann equation as continuous drift–diffusion phenomena in velocity space. Consistency between the FP method and the Boltzmann equation is achieved by matching production terms. The Maxwell molecular model has been widely used in this process due to the possibility of obtaining closed-form solutions for these production terms. However, it is well known that the Maxwell molecular model has difficulty predicting strong shock waves since it cannot provide accurate relaxation rates for the moments. By contrast, the variable hard-sphere (VHS) molecular model is able to capture the transport properties of real gases better than the Maxwell molecular model. Nonetheless, there have so far been no reports associated with an accurate VHS molecular model for the stochastic particle FP method. In this paper, two different molecular models are developed to describe a monatomic gas interacting through a VHS potential. The proposed VHS molecular models are evaluated using Grad's 13- and 26-moment distribution functions; hence, they are named the G13 and G26 molecular models. The G13 and G26 molecular models include additional nonlinear moments compared with the conventional Maxwell molecular model. A one-dimensional shock wave and two-dimensional hypersonic cylinder flow are considered for validation. The results show that the proposed molecular models perform better than the Maxwell molecular model in predicting supersonic and hypersonic shock waves.