An implicit unified gas-kinetic scheme for unsteady flow in all Knudsen regimes

Abstract

The unified gas-kinetic scheme (UGKS) is a direct modeling method for multiple scale transport. Based on the ratio of time step to the particle collision time, the local evolution solution on the mesh size and time step scales is used in the construction of the multiscale method. For a flow problem covering multiple flow regimes, such as the hypersonic flow around a flying vehicle in near space, the UGKS is able to capture the highly compressed Navier-Stokes solution in one region and fully expanded free molecular flow in another region, with significant variations of the ratio between the time step and the local particle collision time around the vehicle. For an explicit UGKS, the time step in the whole computational domain is determined by the CFL condition. With implicit and multigrid techniques, the efficiency of the UGKS [1,2] has been improved by two orders of magnitude for steady state computation. However, for unsteady flow computation, due to the CFL condition the global time step used in the explicit UGKS may be limited by the smallest cell size in the computational domain. As a result, for a largely stretched non-uniform mesh the global time step becomes very small and the ratio of the time step to the local particle collision time may get a very small value. Under such a circumstance, even though the UGKS is a multiscale method, the real physics represented in the explicit UGKS may be constrained to the kinetic scale transport only, and the advantage of the multiscale nature in UGKS has not been fully utilized. In order to solve the multiscale unsteady flow problem efficiently, the time step restriction from a global CFL condition has to be released. In this paper, we will develop an implicit UGKS (IUGKS) for unsteady flows by alternatively solving the macroscopic and microscopic governing equations within a time step iteratively. With a pre-defined uniform large evolution time step, the local CFL number varies greatly in different region, such as on the order 1 in the large numerical cell size region, and 100 in the small cell size region. In order to preserve coherent flow evolution and keep the multiscale nature, the time averaged numerical flux across a cell interface is still evaluated by the explicit UGKS under the local CFL condition. Therefore, the multiscale property of the UGKS modeling has been kept over non-uniform meshes. With improved temporal discretization, the current IUGKS can automatically go back to the explicit UGKS and obtain identical solutions when the time step of the implicit scheme gets to that of an explicit one. Many numerical examples are included to validate the scheme for both continuum and rarefied flows with a large variation of artificially generated mesh size. The IUGKS has a second order accuracy and presents reasonably good results for unsteady flow computation, and its efficiency has been improved by dozens of times in comparison with the explicit UGKS.