Accuracy of Bhatnagaar-Gross-Krook Scheme in Solving Laminar Viscous Flow Problems

Abstract

This paper describes the development of a gas-kinetic solver to compute laminar viscous flows in two-space dimensions via a finite difference approach. The convection flux terms of the Navier-Stokes equations are discretized by a semidiscrete finite difference method. The resulting inviscid flux function is then determined by a numerical scheme that is based on the Bhatnagaar-Gross-Krook model of the approximate collisional Boltzmann equation. The scheme is based on the direct splitting of the inviscid flux function with inclusion of particle collisions in the transport process. As for the diffusion flux terms, they are discretized by a second-order central difference scheme. The cell interface values required by the gas-kinetic scheme are reconstructed to higher-order spatial accuracy via the monotone upstream-centered schemes for conservation laws variable interpolation method. An explicit-type time integration method known as the modified fourth-order Runge-Kutta is employed for computing steady-state solutions. In the numerical case studies, the results obtained from the flux vector splitting Bhatnagaar-Gross-Krook scheme are compared with available experimental data, analytical solutions, the results from upwind schemes, and the results from central difference scheme to verify the accuracy and robustness of the gas-kinetic solver. The tests have shown that the Bhatnagaar-Gross-Krook scheme is able to resolve the shear layer, the shock structure, and the flow accurately as the results compare favorably with the available experimental and analytical data.