First Investigations on High-Order Discretization Schemes for Finite Volume Methods in Cartesian Grids

Abstract

Based on a finite-volume method, a spatially fourth-order accurate scheme is proposed for solving the unsteady compressible Euler Equations for subsonic flows on Cartesian background grids in the frame of a Chimera method (Ref [1], [2]). The Chimera grids serve as background grids while curvilinear grids are used close to the solid surface of an airfoil or wing. The basic idea to achieve the high-order accuracy is realized by defining the flow values or flux terms at the interfaces of the control cell with high-order interpolation. Accuracy analysis of the high order discretization schemes is given and compared with the conventional second-order discretization scheme. A dual-time stepping method is used for time integration, thus making these schemes best suited for unsteady problems. To verify the improvement of the fourth-order scheme, a two-dimensional convecting vortex in a uniform stream and 2D BVI effects are simulated. For the 2D BVI simulations, a curvilinear grid with the original second-order scheme is used as a child grid to deal with the realistic solid surface and flow discontinuities in the near flow region. The fourth-order scheme is applied on the Cartesian background grid. The results of the simulations show that the diffusion of the moving vortex is considerably reduced compared to the original second-order discretization scheme.