Wavelet analysis of 2D turbulent fields

Abstract

Two descriptions of turbulent two-dimensional incompressible flows are at hand: the à la Kolmogorov spectral scenario is based on non-linear interactions between scales and cascade processes, when the coherent structures description deals with spatially localized interacting structures. The spectral approach leads to universal spectral or dispersion laws which often fail to be verified as this approach does not take into account the strong spatial variability of the flow. Conversely the coherent structure description does not provide quantitative predictions. From the Lagrangian point of view the two types of description have led to different approaches for the study of turbulent transport; namely, global self-similar theories and local parameterizations of turbulent dispersion. To reconcile these two kind of approaches tentatives have been made to separate flow fields into regions having different dynamical behaviours. However to achieve this program we need tools to obtain quantitative information on the dynamical properties of these different regions. To this end, orthonormal multidimensional wavelet decompositions are presented on a theoretical and practical basis as a preliminary step towards the goal of properly parameterizing Eulerian and Lagrangian properties of turbulent flows. Wavelet coefficients are interpreted from the viewpoint of turbulence analysis and used to extract form the flow field spatially localized spectral informations, giving a simultaneous description of the flow field in physical space (vortices and structures, intermittency) and in spectral space (energy or enstrophy spectra and cascades). Applications of the wavelet analysis are given for two-dimensional flows exhibiting a strong spatial variability; one is related to an unforced temporal mixing layer, the other to forced homogeneous incompressible turbulence. In both cases, the wavelet analysis points out clearly that small scales are dominant in shear zones, outside large scale vortices and that local spectral slopes are much different within a large scale coherent vortex, a shear zone, or the intersticial flow.