A multilevel method applied in the nonhomogeneous direction of the channel flow problem

Abstract

In this paper, we apply a spectral multilevel method in the nonhomogeneous direction of a channel. The spectral tau method being not well suited to separate the scales, we use a Galerkin basis in the wall normal direction. Then we can separate the scales, as in the periodic case, from the spectral decomposition of the velocity field. In this way, the quantities associated with the small and large scales verify the no slip boundary conditions. Then, we resolve the large and the small scale equations, simplifying the computation of the small scales. Indeed, we use a quasi-static approximation to compute the small scales. As for the interaction terms, we apply a quasi-static approximation to estimate the modulus, the phase being updated, at each time step, in function of the large scales. To validate the method proposed, we have done two simulations of the channel with the multilevel method. They correspond to two different choices of the total number of modes and of the coarse cut-off level for the multilevel method in the wall normal direction. The results obtained are compared with the results stemming from direct numerical simulations (DNS): one fine DNS (fine resolution) and one low DNS (coarse resolution).