A Hybrid Finite Difference-WENO Scheme for Large Eddy Simulation of Compressible Flows

Abstract

For a wide range of applications, a better understanding of the physical mechanisms involved during the development of hydrodynamic instabilities at interfaces separating two gases is desired. We are interested in flows that contain both turbulence and shock waves, and that are characterized by a high Reynolds number. Current computational power does not allow direct numerical simulation of these flows and then only a large eddy simulation (LES) is possible. This work aims at developing a numerical scheme for compressible LES. Turbulence is better simulated when the numerical method is not dissipative and a high-order of accuracy is recommended for under-resolved simulations. However, the presence of shocks implies the use of a shock-capturing type scheme to stabilize the solution, with the risk of (eventually) overwhelming the small scales of the solution. A compromise can be obtained by coupling a high-order central finite difference scheme for turbulent regions with a WENO scheme for discontinuous ones. To reduce instabilities in non-dissipative LES of turbulent flows, the convective operators are expressed using a skew-symmetric like form. The semi-discretized problem in space should be carefully constructed to ensure the local preservation of kinetic energy by convection. By considering a symmetric stencils collection for the WENO flux reconstruction, the final hybrid flux can be expressed as the sum of the non-dissipative term and a dissipative one. The solution smoothness is estimated component-wise during the WENO reconstruction in the local characteristic space, to avoid the use of excessive numerical dissipation.