Adaptive Partitioning-based Discrete Unified Gas-Kinetic Scheme for Flows in All Flow Regimes

Abstract

The discrete unified gas kinetic scheme (DUGKS) is a multiscale approach, which can be used to obtain reasonable results in all flow regimes. The key of this method is the reconstruction of numerical fluxes at the cell interface by coupling the motion of particles from their collisions, namely the use of the discrete characteristic solution to the Boltzmann-BGK equation at the cell interface to calculate numerical fluxes. But like all the discrete velocity methods (DVMs), the computational cost of DUGKS is determined by the discretization in both the physical space and the velocity space. For the continuous flow region in the computational domain, the discretization in the velocity space is unnecessary since the distribution function can be reconstructed from the Chapman-Enskog expansion directly. To improve the efficiency of DUGKS in capturing cross-scale flow physics, an adaptive partitioning-based discrete unified gas kinetic scheme (ADUGKS) is developed in this work. The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation, which contains the initial distribution function and the local equilibrium state. The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes. If the contribution of the initial distribution function is negative., the local flow field can be regarded as the continuous flow and the Navier-Stokes (N-S) equations can be used to obtain the solution directly. Otherwise, the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefied effect. Given this, the computational domain is divided into the DUGKS cell and the N-S cell based on the contribution of the initial distribution function to the calculation of free streaming fluxes. In the N-S cell, the local flow field is evolved by solving the Navier-Stokes equations, while in the DUGKS cell, both the discrete velocity Boltzmann equation and the corresponding macroscopic governing equations are solved by a modified DUGKS. Since more and more cells turn into the N-S cell with the decrease of the Knudsen number, a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.