Evaluation of stochastic particle bhatnagar-gross-krook method based on velocity distribution function and variance-reduced method for low-speed flows

Abstract

The Boltzmann equation and the Direct Simulation Monte-Carlo (DSMC) method, a stochastic approach to solving it, are used to analyze a variety of non-equilibrium flows, including rarefied flows, supersonic flows, and microscale flows. However, the efficiency of the DSMC method decreases rapidly in near-equilibrium flows where the number of collisions between particles increases. As an alternative, the stochastic particle Bhatnagar-Gross-Krook method has been studied. This work addresses two limitations of the stochastic particle BGK method. The first is the limitation in evaluating the accuracy. Various BGK models, such as ellipsoidal statistical BGK and Shakhov BGK, have been proposed to reproduce DSMC. However, previous studies analyzing the accuracy of each model in non-equilibrium flows only evaluated the accuracy of macroscopic variables such as density, velocity, and temperature. Since the velocity distribution function (VDF) deviates from the Maxwellian distribution in non-equilibrium flows, a direct comparison of the VDF is more appropriate for the accuracy analysis of stochastic particle BGK methods. Therefore, in this study, the VDFs of two stochastic particle BGK methods in non-equilibrium flows are compared and the differences are quantified to evaluate their accuracy. It is found that in nonequilibrium flows, the accuracy of the moments does not consistently reflect the accuracy of the VDF. The second limitation is the analysis of low-speed flows. Due to statistical uncertainty, stochastic particle BGK methods are inefficient in distinguishing the flow property of interest from statistical noise in slow flows. In this study, we develop and validate an efficient variance-reduced BGK method for slow flows using the importance sampling technique. It is shown that the variance-reduced BGK method can be used to efficiently solve low-speed flows with minimal gradients in density, velocity, and temperature.