Vertex corrections and 'optimal' subgrid models for homogeneous isotropic turbulence

Abstract

If the Fourier modes in a DNS of homogeneous isotropic turbulence are separated into 'resolved' (k < kc) and 'unresolved' (k > kc) modes by introducing a partition wavevector kc, projection of the part of the nonlinear interaction containing unresolved modes onto the resolved velocity gives a numerical eddy viscosity that models the effect of the unresolved interactions on the resolved velocity field. Evaluation of this eddy viscosity by DIA gives a result in fair overall agreement with numerical data. We consider the extension of this formalism to projection of the unresolved nonlinearity onto quadratic products of the resolved velocity. We evaluate this projection by perturbation theory, and attempt to relate the result to the vertex corrections predicted by Martin-Siggia-Rose theory. Non-Gaussian properties of turbulence prove to have a crucial role. We discuss the constraints imposed by Galilean invariance on this type of computation.