The turbulent energy cascade | Index |

From November 1999 until February 2002 I have been working on the DFG-funded project KL-611/10 entitled "Small-scale instabilities and their effect upon the turbulent energy cascade", which was supervised by Rupert Klein.

In a statistical sense, idealized turbulent flows can be approximately
described by the Kolmogorov cascade model and additional corrections
for small-scale intermittency (cf. [9]). Several
possible localized "prototype" structures compatible with a
*k ^{-5/3}* inertial range of the energy spectrum have been proposed in
the past (cf. [10] and references therein).
The detailed mechanism, however, describing the localized inter-scale
flow of energy has so far eluded a theoretical description. A similar
uncertainty prevails regarding the way that energy is dissipated by
the smallest scales.

This project starts with the observation that an ensemble of
locally scale-separated events can lead to a continuous spectrum if
the respective length scales themselves vary
stochastically. Therefore, the individual events should still be
accessible to a multiple scales analysis. In this spirit, we expand
the velocity (and consistently the vorticity) fields *locally* in
terms of a small parameter, and derive a non-linear
equation describing the small-scale vorticity. It contains source
terms which describe the linear interaction between the background
field and the small scales. Interestingly, one expression corresponds
to the stretching of large-scale vorticity by the small-scale field, a
mechanism by which small-scale vorticity can be generated *ab initio* in the presence of irrotational small-scale velocity
perturbations. This scenario seems a strong potential candidate for a
description of the origin of the energy cascade.

In order to further investigate the hypothesis in the light of real flow fields, several routes are open. One consists of performing a local scale separation upon the data formally (e.g. using wavelets) and then check for consistency with the equations obtained by the above asymptotic analysis. More specifically, one is interested in finding and describing real "events" which - as theoretically predicted - correspond to almost irrotational small scales transferring vorticity from larger ones. Most of our present effort goes into this direction.

The tasks required by this objective are the following:

- generate data via direct numerical simulation (DNS),
- find a feasible method of local scale decomposition, i.e. an adequate wavelet basis,
- come up with significant search criteria for the detection of small-scale instabilities.

Two - purposefully - distinct flow configurations were selected, which are described in more detail in the following parts of the document. Furthermore, as indicated above, we have chosen to resort to the wavelet formalism, namely discrete-orthogonal wavelet bases, in order to decompose flow fields with respect to space and scale. Documents regarding the wavelet transform (its application to the closed interval and divergence-free vector fields as well as implementation issues) are also treated in the following.

Our search for candidate events in the sense of the asymptotic theory
has so far not been conclusive. The results up to August 2001 have
been summarized in an intermediate report (part
1,
part 2; in
German).
At a later stage we have slightly
modified our approach and continued investigating the asymptotics in
the light of a more general *ansatz*. The idea of analyzing the
non-linear energy transfer in wavelet space is based on work of
Nakano [12] and Meneveau [13] and has
only recently been revitalized by the development of divergence-free
wavelets by Kishida *et al.* [14][15]. We
are considering the enstrophy equation instead and work on methods
of significantly reducing the number of degrees of freedom in order
to reach a computationally feasible system.

markus.uhlmann AT kit.edu

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