Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method

Our aim in this article is to study a new method for the approximation of the Navier-Stokes equations, and to present and discuss numerical results supporting the method. This method, called the nonlinear Galerkin method, uses nonlinear manifolds which are close to the attractor, while in the usual Galerkin method, we look for solutions in a linear space, i.e., whose components are independent. The equation of the manifold corresponds to an interaction law between small and large eddies and it is derived by asymptotic expansion from the exact equation. We consider here the two- and three-dimensional space periodic cases in the context of a pseudo-spectral discretization of the equation. We notice however that the method applies as well to more general flows, in particular nonhomogeneous flows.