The Nonlinear Galerkin Method: A multi-scale method applied to the simulation of homogeneous turbulent flows
چکیده
82 Conclusions In this article, we have proposed a multilevel method which treats diierently the large and the small scales of homogeneous isotropic ows. Moreover, we have derived mathematical estimates for all the parameters (cut-oo level, time scales) involved in this dynamical procedure leading to a completely self-adaptive procedure. Firstly, several computations have been conducted in the context of DNS, i.e. the whole spectrum up to the dissipative scale was simulated. In such case, the multi-level method is able to recover a velocity eld with the same spatial resolution than the Galerkin method, but with a substantial speed-up of at least 2 in CPU time. Secondly, in an approach similar to LES, we have decreased the number of modes retained for the resolved scales and used the same algorithm to modelize the interaction between the low and high frequencies. In such case, we have seen that when the resolved scales are larger than the dissipative ones, but of the same order of magnitude (1/12 as compared to 1/17), the large eddies of the ow are captured correctly; this is not the case when the pseudo-spectral Galerkin method is used with only 32 2 modes, i.e. with a larger scale of the order of 1/16. Although, we only presented computational for moderately large Reynolds numbers, where the small scales are strongly dependent on the energy-containing eddies, we think that these results are very promising. Indeed, we can reasonably hope that the multilevel (nonlinear Galerkin) method can be used eeciently for Large Eddy Simulations. Results on this point will be presented elsewhere, in the case of two and three dimensional homogeneous isotropic ows. Finally, we want to observe that, by opposition with LES methods, no assumptions on the energy spectra or on the velocity correlations are made here. Therefore the method can be applied to the simulation of non-homogeneous ows. First attempts have been made in this direction, for the channel ow problem. Our eeorts will be concentrated on this problem in the near future.