Implementing the direct relaxation process in the stochastic particle method for flexible molecular collisions

Multi-scale phenomena are prevalent and significant across various disciplines. For multi-scale flow physics in the gas-kinetic theory based on Boltzmann equation or its simplified mathematical models (called Boltzmann model equations), the multi-scale mechanism can be modeled by the philosophy of unified modeling, where the free transport behaviors of gas particles and their collision behaviors are coupled by the temporal integral solutions (or characteristic line solutions) of Boltzmann model equations, which leads to a unified/multi-scale property in all scales. Also, the stochastic particle methods are based on these Boltzmann model equations. The corresponding numerical methods are, thus, limited by these model equations. This paper aims to overcome this restriction by replacing these modeled collision operators with a simple direct relaxation (DR) process. Since the collision term of Boltzmann model equation should fulfill the correct relaxation rates of non-equilibrium macro-variables, such as stress tensor and heat flux vector, along with other basic properties, such as conservation and H theorem, the DR process is designed to be directly based on these crucial relaxation rates. Therefore, with the DR strategy for calculating particle collisions, the numerical method can be established without constructing collision operator. Furthermore, the DR has the flexibility and simplicity to recover various models. In this work, Xu's and Yuan's new models are recovered in to illustrate the validation and performance of DR. Moreover, since at the inlet/outlet boundaries, subsonic, supersonic, and hypersonic flows can simultaneously exist, a generalized numerical boundary condition is also considered in the particle methods developed in this paper. Finally, the validation and accuracy of the present method are examined with a series of test cases.