The Nonlinear Galerkin Method: A multi-scale method applied to the simulation of homogeneous turbulent flows

Using results of DNS in the case of two-dimensional homogeneous isotropic flows, we first analyze in detail the behavior of the small and large scales of Kolmogorov like flows at moderate Reynolds numbers. We derive several estimates on the time variations of the small eddies and the nonlinear interaction terms; those terms play the role of the Reynolds stress tensor in the case of LES. Since the time step of a numerical scheme is determined as a function of the energy-containing eddies of the flow, the variations of the small scales and of the nonlinear interaction terms over one iteration can become negligible by comparison with the accuracy of the computation. Based on this remark, we propose a multilevel scheme which treats differently the small and the large eddies. Using mathematical developments, we derive estimates of all the parameters involved in the algorithm, which then becomes a completely self-adaptive procedure. Finally, we perform realistic simulations of (Kolmorov like) flows over several eddy-turnover times. The results are analyzed in detail and a parametric study of the nonlinear Galerkin method is performed