A major problem occuring in the simulation of turbulence is that the number of degrees of freedom of such flows grows up like a power of the Reynolds number. Hence, Direct Numerical Simulations are impossible for complicated flows and in most of the computations a model is used to take into account the effect of the small scales lying in the dissipation range without computing them explicitly.
Like in multigrid methods for linear systems, we need a differentiated treatment of the low and high frequency components of the flow. In the case of the Navier-Stokes equations, such treatment based on physical modelling is usually proposed in the Large Eddy Simulation approach. Our aim here, in relation with Dynamical Systems Theory and the Nonlinear Galerkin Method is to develop such methods, based on the approximation of attractors and on numerical adaptative procedures.
Based on a decomposition of the unknowns into their small and large scales components, we have shown that most of the energy, as well as the enstrophy, is carried by the large scales. Moreover, the time behavior of those components are different : the time derivative of the small scale is smaller than the time derivative of the large ones. Hence, even if the effects of the small scales are non negligible globally, their order and their variations can be much smaller than the accuracy of the computation. In that case, there is no need to compute them.
We have introduced some criterias which define the proper mesh size and time step for the large scales during the time evolution. The small ones are computed by using a nonlinear manifold (approximate inertial manifold) which is close to the attractor. The equation of this manifold corresponds to an interaction law between the small and the large eddies.
Finally, we obtain a multi-level adaptative scheme combining time and space discretization. This new method allows us to recover the same accuracy as for a classical scheme, but with a speed-up at least equal to 2. The application of the method to homogeneous (space periodic) flows and non-homogeneous (channel) flows will be presented here.