Turbulence spectra characteristics of high order schemes for direct and large eddy simulation

The comparative resolution of the high wavenumber portion of compressible turbulence energy spectrum by some high order numerical schemes is presented in this paper. Included in this are the essentially nonoscillatory (ENO) schemes, the weighted essentially nonoscillatory schemes (WENO), and the compact differencing schemes. The governing equations are the Navier-Stokes equations and the objective is to identify the numerical scheme that best represents the physics of compressible turbulence. Mach numbers M₁ values of 0.1, 0.5 and 0.7 are studied. The compact differencing schemes need filters for numerical stability. It is found in this work that a parameter in the filter scheme provides some flexibility for controlling the physical turbulence energy transfer rate at high wavenumbers, vis-a`-vis the numerical dissipation at those scales. Although the ENO schemes do not require filters for numerical stability, the present study shows that the addition of filters improves the energy transfer process at high wavenumbers. Without filtering, with relatively coarse grids, numerical turbulence caused by stencil adaptation persists. This limits the useful wavenumber resolution range of the ENO schemes. The WENO schemes do not require the stabilizing filters but the results tend to be slightly more dissipative. Finally, at low Mach numbers, the current compact differencing and filter scheme formulation gives better results but as the Mach number increases the relative suitability of the ENO method increases.