A simplified discrete unified gas kinetic scheme for incompressible flow

Abstract

The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows, which combines the benefits of both the lattice Boltzmann method and UGKS. By the reconstruction of the gas distribution function using particle velocity characteristic lines, the flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with the reconstruction stage on a whole time step instead of a half time step in the original DUGKS. Using the temporal/spatial integral Boltzmann Bhatnagar–Gross–Krook equation, the auxiliary distribution function with the inclusion of the collision effect is adopted. The macroscopic and mesoscopic fluxes of the cell on the next time step are predicted by the reconstruction of the auxiliary distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on the next time step can be updated through its flux, which is a moment of the predicted mesoscopic flux at cell interfaces. The equilibrium distribution function on the next time step can also be updated. The gas distribution function is updated by the FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. The numerical time step is only limited by the Courant–Friedrichs–Lewy condition, and a relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, and microcavity flow cases, are conducted to validate the present scheme. The observed numerical simulation results reasonably agree with the reported results.