A two-stage fourth-order gas-kinetic scheme on unstructured hybrid mesh

This paper presents an accurate and robust temporal two-stage spatial fourth-order gas-kinetic scheme (GKS) on two dimensional unstructured hybrid mesh for prediction of incompressible and compressible viscous flows. For generalized Riemann problem and Navier–Stokes solution, the GKS provides a time-accurate flux solver using a different way in the reconstruction at a cell interface. Different from previous one-stage time-stepping method, the two-stage Lax–Wendroff type time stepping method is applied in this paper. Compared to standard four-stage fourth-order Runge–Kutta method, the two-stage fourth-order time accurate method reduces the complexity of time derivative of numerical flux. To achieve fourth-order accuracy, a finite volume method for GKS using cubic spline reconstruction is proposed on both structured and unstructured meshes. When dealing with flow discontinuities, the original piecewise equation for spline scheme is blended with shock-capturing WENO scheme. In resolution of shock wave, adaptive mesh refinement is applied. Plenty of one and two-dimensional test cases, including Taylor–Green vortex flow, Taylor–Couette flow, Shu–Osher problem, Woodward–Colella blast problem, two-dimensional Riemann problem, viscous shock tube flow, supersonic flow over a forward-facing step, and hypersonic flow over a circular cylinder, are carried out to validate and demonstrate performance of the proposed scheme.