On the multidimensional gas-kinetic BGK scheme

This paper concerns the performance difference among the multidimensional (MD) gas-kinetic BGK scheme, the corresponding quasi-one-dimensional (Q1D) extension and the directional splitting (DS) scheme with kinetic boundary conditions. The MD scheme includes tangential slopes in the flux calculation, which are absent in a Q1D or DS method. In spite of taking more computational time, the MD scheme is found to be able to capture the characteristics of rarefied flow having a boundary with curvature or a nonuniform temperature, such as the inverted velocity distribution in rarefied cylindrical Couette flow and the weak flow field induced by the temperature gradient of a body, where the Q1D fails. The MD scheme can also yield clearly better results in high-speed microchannel flow and power-law fluid flow between concentric rotating cylinders, when compared with direct simulation Monte Carlo studies and analytic solutions. In the low-Reynolds-number flow around a NACA0012 airfoil case, the MD scheme and the Q1D method predict wall heat flux distribution with obvious difference. The DS method predicts results nearly identical to those from Q1D in these steady flow cases, except that it can hardly give reasonable solutions in power-law fluid case with viscous exponential factor away from unity. Correct prediction of the stress and the wall temperature gradient is considered responsible for this better performance. In simulations of the scalar convection–diffusion equation, it is found that the inclusion of tangential slopes in the flux computation can clearly improve the temporal accuracy of the scheme. In this case, the DS scheme also shows good performance. The present study suggests caution in adopting of the Q1D extension or DS method in multidimensional flow when it is sensitive to the accuracy of stress or wall variable gradient calculations. It is better to use the MD scheme for high-performance simulation.