Efficient calculations of a gas-kinetic BGK scheme for inviscid and viscous steady flows

Abstract

This paper presents the efficient calculations of the finite volume gas-kinetic BGK scheme for inviscid and viscous steady flowfields. As the first step toward the efficient calculation of steady state problems, particle distribution functions in the solutions of the BGK model are simplified to the extent that the essential features of the original gas-kinetic BGK scheme are not lost. Convergence acceleration techniques.as local time stepping and multigrid are developed for the present scheme in the framework of an implicit time integration. Noticing a few difficulties that may arise in compressible viscous flows, two variant schemes are proposed for the purpose of an exact resolution of contact discontinuity. Along with the development of two variant schemes, a Pr number correction method is developed to allow the present schemes to work for arbitrary pr number. The present schemes with some modifications and convergence acceleration techniques are applied to various steady state problems ranging from inviscid to viscous turbulent flows to give promising results. ._ L INTRODUCTION As a way to predict the compressible flows by using the ability of fast computing machines, many numerical schemes have been developed. ‘Among them, the Riemann solver, exact or approximate, and flux vector splitting are notable and are widely applied for various.aerodynamic problems with a varying degree of success. They attempt to resolve the wave interaction phenomena of the Euler equations in an upwind manner. For viscous calculation, the centrally differenced viscous flux is explicitly added to the inviscid flux. As those numerical methods are applied to, yarious aerodynamic problems, some serious drawbacks also surfaced. Flux vectbr splitting schemes generally .give v&y efficient and robust results even in severe environments, but the ignorance of the contact discontinuities of the linear wave precludes an exact calculation of viscous flows, which is not alleviated merely by a higher order spatial accuracy. The most popular scheme today in a family of Riemann solvers is known to be the Roe’s FDS’(Flux Difference Splitting), which enables us to ,capture compressible tlow phenomena quite accurately. Quirk’, however, pointed out that the robustness and accuracy of Roe’s method are severely inflicted from the even-odd decoupling, carbuncle phenomenon, and difficulties near low density flows. These shortcomings not only deteriorate the accuracy of solutiqn,. but, in some situations, prevent us from obtaining the solution itself. Based on wholly different physics dealing with the statistical behaviors of particles via the Boltzmann equation, several gas-kinetic numerical schemes have been independently developed by Reitz3, Perthame4, Pullin’, and Deshpande’. They designed, in some. manner, a upwinding scheme in particle level from the collisionless Boltzmann equation. Since their methods allow particles to penetrate-a cell interface without collisions, they usually produce a large numerical viscosity and heat conductivity, whose merit and demerit are shared with flux vector splitting schemes from the Euler equations. One of the interesting approaches taking a collision effect into consideration-can be found in Xu and Prendergast. 7*8 In this method, the collision effect is considered by the BGK model as an approximation of the collision integral in the BoltzmFn equation. It is found that a gas-kinetic BGK scheme has several advantages over Riemann solvers and many desirable properties numerically and physically. This paper notices the difficulties of a BGK scheme that may arise in compressible viscous calculations and focuses on its remedy and the development of new BGK-type numerical schemes with convergence acceleration techniques and Pr number correction . This paper is organized as follows. In Se&ion II, the construction of the standard gas-kinetic BGK scheme is presented. Referring to a few issues in cbmpressible viscous calculations, two new variant schemes with new features * Graduate Research Assistant, Dep’t of Aerospace Engineering, Student Member AIAA. ’ Assistant Professor, Dep’t of Aerospace Engineering, Member AIAA. J Professor, Dep’t of Aerospace Engineering, Senior Member AIAA. Copyright @ 1999 by the American Instiitute of Aeronautics and Astronautics, Inc. All rights reserved.