An improved unified gas-kinetic scheme and the study of hock structures

With discretized particle velocity space, a unified gas-kinetic scheme for entire Knudsen number flows has been constructed based on the Bhatnagar-Gross-Krook (BGK) model (2010. J. Comput. Phys., 229, 7747-7764). In comparison with many existing kinetic schemes for the Boltzmann equation, besides accurate capturing of non-equilibrium flows, the unified method has no difficulty to get accurate solution in the continuum flow regime as well, such as the solution of the Navier-Stokes equations. More importantly, in the continuum flow regime, the time step used by the unified scheme is determined by the Courant-Friedrichs-Lewy condition, which can be many orders larger than the particle collision time. In some sense, the unified method overcomes the time step barrier for many kinetic methods, such as direct simulation Monte Carlo (DSMC), direct Boltzmann solver and many other kinetic solvers. The unified scheme is a multiscale method, where the macroscopic flow variables and microscopic gas distribution function are updated simultaneously. In an early approach in the unified BGK (U-BGK) scheme, the heat flux in the BGK model is modified through the update of macroscopic flow variables, then this modification feeds back into the update of non-equilibrium gas distribution function. In this paper, we are going to develop a unified scheme for the Shakhov model, the so-called unified Shakhov (U-Shk), where the heat flux is corrected directly through the modification of gas distribution function. Theoretically, it will be shown that current U-Shk is more consistent than the U-BGK for the highly non-equilibrium flow computations. The study of shock structures from low to high Mach numbers will be presented and the simulation results will be compared with DSMC solutions as well as possible experimental measurements. The result improvement of U-Shk over U-BGK is clearly achieved. Based on the simulation results, now we fully believe that the unified scheme is an accurate and efficient flow solver in all Knudsen number flow regime.