Numerical investigation from rarefied flow to continuum by solving the Boltzmann model equation

Based on the Bhatnagar-Gross-Krook (BGK) Boltzmann model equation, the unified simplified velocity distribution function equation adapted to various flow regimes can be presented. The reduced velocity distribution functions and the discrete velocity ordinate method are developed and applied to remove the velocity space dependency of the distribution function, and then the distribution function equations will be cast into hyperbolic conservation laws form with non-linear source terms. Based on the unsteady time-splitting technique and the non-oscillatory, containing no free parameters, and dissipative (NND) finite-difference method, the gas kinetic finite-difference second-order scheme is constructed for the computation of the discrete velocity distribution functions. The discrete velocity numerical quadrature methods are developed to evaluate the macroscopic flow parameters at each point in the physical space. As a result, a unified simplified gas kinetic algorithm for the gas dynamical problems from various flow regimes is developed. To test the reliability of the present numerical method, the one-dimensional shock-tube problems and the flows past two-dimensional circular cylinder with various Knudsen numbers are simulated. The computations of the related flows indicate that both high resolution of the flow fields and good qualitative agreement with the theoretical, DSMC and experimental results can be obtained.