Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence
using an adaptive orthogonal wavelet basis
  Marie FARGE, Kai SCHNEIDER and Nicholas KEVLAHAN

M.F.: Laboratoire de Météorologie Dynamique, École Normale Supérieure, 24, rue Lhomond, 75231 Paris cedex 05, France
and Centre de Mathématiques et Leurs Applications, Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex, France
Institut für Chemische Technik, Universität Karlsruhe (TH), Kaiserstrasse 12, 76128 Karlsruhe, Germany;
N.K.: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
and Centre de Mathématiques et Leurs Applications, Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex, France

Abstract We decompose turbulent flows into two orthogonal parts : a coherent, inhomogeneous, non-Gaussian component and an incoherent, homogeneous, Gaussian component. The two components have different probability distributions and different correlations, hence different scaling laws. This separation into coherent vortices and incoherent background flow is done for each flow realization before averaging the results and calculating the next time step. To perform this decomposition we have developed a nonlinear scheme based on an objective threshold defined in terms of the wavelet coefficients of the vorticity. Results illustrate the efficiency of this coherent vortex extraction algorithm. As an example we show that in a 2562 computation 0.7% of the modes correspond to the coherent vortices responsible for 99.2% of the energy and 94% of the enstrophy. We also present a detailed analysis of the nonlinear term, split into coherent and incoherent components, and compare it with the classical separation, e.g., used for large eddy simulation, into large scale and small scale components. We then propose a new method, called cherent vortex simulation (CVS), designed to compute and model two-dimensional turbulent flows using the previous wavelet decomposition at each time step. This method combines both deterministic and statistical approaches : (i) Since the coherent vortices are out of statistical equilibrium, they are computed deterministically in a wavelet basis which is remapped at each time step in order to follow their nonlinear motions. (ii) Since the incoherent background flow is homogeneous and in statistical equilibrium, the classical theory of homogeneous turbulence is valid there and we model statistically the effect of the incoherent background on the coherent vortices. To illustrate the CVS method we apply it to compute a two-dimensional turbulent mixing layer.

Non Gaussianité et simulation des tourbillons cohérents en base d'ondelettes adaptative
pour la turbulence bidimensionnelle
Résumé Cet article montre qu'à l'aide des ondelettes on peut décomposer les écoulements turbulents en deux types : le premier non Gaussien, qui correspond aux tourbillons cohérents, et le second Gaussien, qui correspond à l'écoulement résiduel produit par les interactions non linéaires entre les tourbillons.