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Non-Gaussianity and
coherent vortex simulation for two-dimensional turbulence
using an adaptive orthogonal wavelet basis
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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
K.S.: 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
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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.
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Non Gaussianité
et simulation des tourbillons cohérents en base d'ondelettes adaptative
pour la turbulence bidimensionnelle
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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. |