The proper orthogonal decomposition (POD), also called Karhunen-Loe`ve expansion, which extracts ‘coherent structures’ from experimental data, is a very efficient tool for analyzing and modeling turbulent flows. It has been shown that it converges faster than any other expansion in terms of kinetic energy (Lumley 1970). First, the POD is applied to the chaotic solution of the Lorenz equations. The dynamics of the Lorenz attractor is reconstructed by only the first three POD modes. In the second part of this paper, we show how the POD can be used in turbulence modeling. The particular case studied is the wall region of a turbulent boundary layer. In this flow, the velocity field is expanded into POD modes in the normal direction and Fourier modes in the streamwise and spanwise directions. Dynamical systems are obtained by Galerkin projections of the Navier Stokes equations onto the different modes. Aubry et al. (1988) applied the technique to derive and study a ten dimensional representation which reproduced qualitatively the bursting event experimentally observed. It is shown that streamwise modes, absent in Aubry et al.’s model, participate to the bursting events. This agrees remarkably well with experimental observations. In both examples, the dynamics of the original system is very well recovered from the contribution of only a few modes.
On the Statistical Stability of Families of Attracting Sets and the Contracting Lorenz Attractor
