The Bgk and Lrs Schemes for Computing Euler and Navier Stokes Flows

Upwind schemes for the Euler and Navier-Stokes equations may be broadly classified into two distinct classes: those based directly on the macroscopic Navier-Stokes equations and those based indirectly on the Navier-Stokes equations via the Boltzmann equation. The former, conventional approach represents the majority of current research. However, recently gas-kinetic methods have gained attention as a robust alternative to the more conventional Navier-Stokes based ones. This work attempts to build on these approaches and see if some improvements may be obtained in schemes of the former class by incorporating features of the latter. The following issues have been addressed: BGK Scheme: The finite volume BGK scheme proposed by Xu [1] is among the most successful gas-kinetic schemes for Navier-Stokes equations. Some lacunae and inconsistencies which preclude its use directly into an industrial strength Navier-Stokes solver can however be identified. An attempt is made here to address some of these issues: 1. The scheme, as proposed restricts the accuracy in time to second order. A series of schemes that remove this restriction are derived. 2. It is well known that the Chapman-Enskog expansion of the BGK equation fixes the Prandtl number of the resultant Navier-Stokes equations at unity. The gas-kinetic BGK method inherits this problem. A simple and inexpensive procedure that corrects this problem is proposed. Unlike previously published fixes, which operate at the level of the numerics, this operates directly at the level of the partial differential equations. This makes it more physically justified apart from being significantly cheaper. 3. The resultant viscous discretization is inaccurate for highly structured grids. An inexpensive correction is proposed which removes this inaccuracy. Further, it is examined if a conventional robust, mesh-transparent viscous discretization may be obtained by