A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow

This paper concerns the development and application of a gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK) model for the Navier-Stokes equations in the study of hypersonic viscous flow. Firstly, we extend the gas-kinetic Navier-Stokes solver [K. Xu, A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys. 171 (2001) 289-335] by implementing a multidimensional particle propagation mechanism in the flux evaluation, where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included. With the construction of a time averaged flux function, an implicit BGK scheme with LU-SGS method is constructed. The main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh, where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation. In the numerical parts, we concentrate on the computation of heat flux in laminar hypersonic viscous flows, where complicated flow phenomena, i.e., shock boundary layer interaction, flow separation, and viscous/inviscid interaction, will be encountered. The cases studied include the type IV shock-shock interaction around a circular cylinder and hypersonic flow passing through a double-cone geometry. In the hypersonic viscous flow, in comparison with the capturing of velocity and pressure fields, the accurate computation of stress and heat flux bears large difficulties. In all cases studied here, the heat fluxes obtained across body surfaces have good agreement with the experimental measurements.