A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows

Abstract

Compressible flow has intrinsically multiple scale nature due to the large variations of gas density and characteristic scale of flow structure, especially in hypersonic and reentry problems. It is challenging to construct an accurate and efficient numerical algorithm to capture non-equilibrium flow physics across different regimes. In this paper, a unified gas kinetic scheme with adaptive velocity space (AUGKS) for multiscale flow transport will be developed. In near-equilibrium flow region, particle distribution function is close to the Chapman-Enskog expansion and can be formulated analytically, where only macroscopic conservative flow variables are updated. With the emergence of non-equilibrium effect, the AUGKS automatically switches to the original unified gas kinetic scheme (UGKS) with a discrete velocity space to follow the evolution of particle distribution function. A criterion is proposed to quantify the non-equilibrium and is used for the switch between continuous and discrete particle velocity space. Following the scale-dependent local evolving solution, the AUGKS presents the discretized gas dynamic equations directly on the cell size and time step scales, i.e., the so-called direct modeling method. As a result, the scheme is able to capture the cross-scale flow physics from particle transport to hydrodynamic wave propagation, and provides a continuous variation of solutions from the Boltzmann to the Euler equations. Different from conventional DSMC-Fluid hybrid method, the AUGKS does not need a buffer zone to match up kinetic and hydrodynamic solutions. Instead, continuous and discrete particle velocity spaces are automatically and robustly switched, such as translating the continuous Chapman-Enskog distribution function to the discrete grid points in the velocity space. Therefore, the AUGKS is feasible for the numerical simulations with unsteadiness and complex geometries. Compared with the asymptotic preserving (AP) method which solves kinetic equation uniformly over entire computational domain with discretized velocity space, the current velocity-space adaptive unified scheme speeds up the computation, reduces the memory requirement significantly, and maintains the equivalent accuracy for multiscale flow simulations. Many test cases validate the current approach. The AUGKS provides an effective tool for non-equilibrium flow studies.