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Small-scale instabilities and their relation to the turbulent energy cascade

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:

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

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