Dispersion-Dissipation Condition for Finite Difference Schemes

A general dispersion-dissipation condition for finite difference schemes is derived by analyzing the numerical dispersion and dissipation of explicit finite difference schemes. The proper dissipation required to damp spurious high wave-number waves in the solution is determined from a physically motivated relation between group velocity and dissipation rate. The application to a previously developed low-dissipation weighted essentially non-oscillatory scheme (WENO-CU6-M2) demonstrates that this condition can serve as a general guideline for optimizing the dispersion and dissipation of linear and non-linear finite difference schemes. Moreover, the improved WENO-CU6-M2 scheme which satisfies the dispersion-dissipation condition can be used for under-resolved simulations. This capability is demonstrated by considering transition to turbulence and self-similar energy decay of the three-dimensional Taylor-Green vortex. Simulations of the inviscid and the viscous Taylor-Green vortex at Reynolds numbers ranging from Re=400 to Re=3000 show a significant improvement over the classical dynamic Smagorinsky model and demonstrate competitiveness with state-of-the-art implicit LES models, while preserving shock-capturing properties.