Fluid dynamic equations are valid in their respective modeling scales. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. In order to study multiscale flow evolution efficiently,
the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region where it is needed. The central ingredient of the UGKS is the coupled treatment of particle transport and collision in the flux evaluation across a cell interface, where a continuous flow dynamics from kinetic to hydrodynamic scales is modeled. The newly developed UGKS has the asymptotic preserving (AP) property of recovering the NS solutions in the continuum flow regime, and the full Boltzmann solution in the rarefied regime. In the mostly unexplored transition regime, the UGKS itself provides a valuable tool for the flow study in this regime. The mathematical properties of the scheme, such as stability, accuracy, and the asymptotic preserving, will be analyzed in this paper as well.
Due to the limited cell resolution in the representation of flow variables, a piecewise continuous initial reconstruction with discontinuous jump at a cell interface is usually used in modern computational fluid dynamics methods. Starting from the di
scontinuity, a Riemann problem in the Godunov method is solved for the flux evaluation across the cell interface in a finite volume scheme. With the increasing of Mach number in the CFD simulations, the adaptation of the Riemann solver seems introduce intrinsically a mechanism to develop instabilities in strong shock regions. Theoretically, the Riemann solution of the Euler equations are based on the equilibrium assumption, which may not be valid in the non-equilibrium shock layer. In order to clarify the flow physics from a discontinuity, the unsteady flow behavior of one-dimensional contact and shock wave is studied on a time scale of (0~10000) times of the particle collision time. In the study of the non-equilibrium flow behavior from a discontinuity, the collision-less Boltzmann equation is first used for the time scale within one particle collision time, then the direct simulation Monte Carlo (DSMC) method will be adapted to get the further evolution solution. The transition from the free particle transport to the dissipative Navier-Stokes (NS) solutions are obtained as an increasing of time. The exact Riemann solution becomes a limiting solution with infinite number of particle collisions. For the high Mach number flow simulations, the points in the shock transition region, even though the region is enlarged numerically to the mesh size, should be considered as the points inside a highly non-equilibrium shock layer.
