Development of Gas-Kinetic BGK Scheme for Two- Dimensional Compressible Inviscid Flows

In this paper, a gas-kinetic scheme based on the BGK (Bhatnagar-Gross-Krook) model is developed for the two-dimensional compressible inviscid flow fields. BGK scheme is an approximate Riemann solver that uses the collisional Boltzmann equation as the governing equation for flow evolutions. For efficient computations, particle distribution functions in the general solution of the BGK model are simplified and used for the flow simulations. High order accuracy is achieved via the reconstruction of flow variables using the MUSCL (Monotone Upstream-Centered Schemes for Conservation Laws) interpolation technique. For steady state problems, an implicit time integration method is adopted here which is more preferable to a multi-stage Runge-Kutta method. To investigate the computational characteristics of BGK scheme in detail, it has been applied to two typical two-dimensional numerical experiments namely, a supersonic wedge and a channel with ramp. The results, compared with some typical schemes like those Steger-Warming (SW) FVS (Flux Vector Splitting), and Roe FDS (Flux Difference Splitting) and available exact solutions are summarized as follows; robust, accurate and high resolution at discontinuities such as shock waves.

[1]  Kun Xu,et al.  A gas-kinetic BGK scheme for the compressible Navier-Stokes equations , 2000 .

[2]  Kun Xu,et al.  A gas-kinetic method for hyperbolic-elliptic equations and its application in two-phase fluid flow , 1999 .

[3]  K. Xu,et al.  Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis , 1999 .

[4]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[5]  Kun Xu,et al.  Gas-Kinetic Theory-Based Flux Splitting Method for Ideal Magnetohydrodynamics , 1999 .

[6]  Shamsul Qamar,et al.  A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations , 2004, SIAM J. Sci. Comput..

[7]  Kun Xu,et al.  Entropy analysis of kinetic flux vector splitting schemes for the compressible Euler equations , 2001 .

[8]  James E. A. John.  Gas Dynamics, Second Edition , 1984 .

[9]  K. Hoffmann,et al.  Computational Fluid Dynamics for Engineers , 1989 .

[10]  Waqar Asrar,et al.  Evaluation of gas-kinetic schemes for solving 1d inviscid compressible flow problems , 2003, Int. J. Comput. Eng. Sci..

[11]  Oh-Hyun Rho,et al.  Development of an Improved Gas-Kinetic BGK Scheme for Inviscid and Viscous Flows , 2000 .

[12]  C. Chow,et al.  Grid adaptation using chimera composite overlapping meshes , 1994 .

[13]  Kun Xu,et al.  Unsplitting BGK-type schemes for the shallow water equations , 1999 .

[14]  Antony Jameson,et al.  GAS-KINETIC BGK METHOD FOR THREE-DIMENSIONAL COMPRESSIBLE FLOWS , 2002 .

[15]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .