Revisiting Numerical Errors in Direct and Large Eddy Simulations of Turbulence: Physical and Spectral Spaces Analysis

Some recent studies on the effects of truncation and aliasing errors on the large eddy simulation (LES) of turbulent flows via the concept of modified wave number are revisited. It is shown that all the results obtained for nonlinear partial differential equations projected and advanced in time in spectral space are not straightforwardly applicable to physical space calculations due to the nonequivalence by Fourier transform of spectral aliasing errors and numerical errors on a set of grid points in physical space. The consequences of spectral static aliasing errors on a set of grid points are analyzed in one dimension of space for quadratic products and their derivatives. The dynamical process that results through time stepping is illustrated on the Burgers equation. A method based on midpoint interpolation is proposed to remove in physical space the static grid point errors involved in divergence forms. It is compared to the sharp filtering technique on finer grids suggested by previous authors. Global performances resulting from combination of static aliasing errors and truncation errors are then discussed for all classical forms of the convective terms in Navier–Stokes equations. Some analytical results previously obtained on the relative magnitude of subgrid scale terms and numerical errors are confirmed with 3D realistic random fields. The physical space dynamical behavior and the stability of typical associations of numerical schemes and forms of nonlinear terms are finally evaluated on the LES of self-decaying homogeneous isotropic turbulence. It is shown that the convective form (if conservative properties are not strictly required) associated with highly resolving compact finite difference schemes provides the best compromise, which is nearly equivalent to dealiased pseudo-spectral calculations.