Marie Farge (Ecole Normale Superieure) and  Kai Schneider (Univesite de Provence)

Wavelet methods to analyze and compute turbulent flows

AbstractTurbulence is characterized by its nonlinear and multiscale behaviour,
self-organization into coherent structures and a generic randomness.
The number of active spatial and temporal scales involved increases
with the Reynolds number, therefore it soon become prohibitive for direct
numerical simulation. However, observations show that for a given flow
realization these scales are not homogeneously distributed, neither in
space nor in time, which corresponds to the flow intermittency.
To be able to benefit from this property, a suitable representation of the
flow should reflect the lacunarity of the fine scale activity, in both
space and time.

A prominent tool for multiscale decompositions are wavelets. A wavelet is
a well localized oscillating smooth function, i.e. a wave packet, which is
dilated and translated. The thus obtained wavelet family allows to
decompose a flow field into scale-space contributions from which it can be
perfectly reconstructed. Note that for finer scales the physical support
of the basis functions is decreasing. The flow intermittency is reflected
in the sparsity of the wavelet representation, i.e. only few coefficients,
the strongest ones, are necessary to represent the dynamically active part
of the flow.

The Coherent Vortex Simulation (CVS) approach we have proposed is based on
the wavelet filtered Navier-Stokes equations. At each time step the
turbulent flow is split into two orthogonal parts, one corresponding to
coherent vortices which are kept, and the other to an incoherent
flow which is discarded.

In the talk we will present first applications of the CVS filter to data
computed by Direct Numerical Simulation (DNS) at high resolution (up to
2048^3 grid points). We will show that the coherent flow can be
represented by few wavelet modes only, which are sufficient to fully
reproduce the vorticity probability density function (PDF) and the energy
spectrum. The discarded incoherent background flow, which is homogeneous,
gaussian and decorrelated, corresponds to the turbulent enstrophy but
has a negligible contribution to the energy.

Finally, we present simulations of a time-developing turbulent mixing
layer where the CVS filter is applied at each time step. The results show
CVS preserves the nonlinear dynamics of the flow, and that discarding the
incoherent modes is sufficient to model turbulent dissipation.

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