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 classes: those based directly on the macroscopic Navier-Stokes equations and those based indirectly on the Navier-Stokes equations via the Boltzmann equation. This work attempts to address two such schemes 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 BGK scheme seems to be a competitive solver for solving viscous flows on unstructured meshes as it has a compact stencil and is mesh independent. Some lacunae which preclude its use into an industrial strength Navier-Stokes solver have been addressed. These include making the scheme amenable to a semi-discrete treatment to enable high order time integration, an inexpensive and physically sound Prandtl number fix, and an important correction in order to get the correct behavior on stretched unstructured meshes. A conventional viscous discretization that mimics BGK's behavior while retaining mesh independence and a compact stencil is proposed. LRS Scheme: An upwinded state scheme is derived. The scheme is seen to be closely related to Roe's scheme which is an upwinded flux scheme but is less expensive in comparison. The scheme is seen to give good results on a range of transonic problems. It also has a qualitatively different behavior from Roe's scheme for a 2D stationary shock subject to perturbations and the hypersonic blunt body problem. The scheme seems to offer another way of attempting to solve the carbuncle phenomenon. SGS modeling and upwinding: A heuristic argument indicating the equivalence of upwinding and SGS modeling in the inviscid limit is presented. The BGK and LRS schemes are tested for decaying compressible turbulence to test certain qualitative aspects of the argument. Corrections required for viscous effects are also discussed.