Multiscale theory of turbulence in wavelet representation

TL;DR

This paper develops a multiscale wavelet-based framework for turbulence in incompressible fluids, enabling divergence cancellation and natural reformulation of Kolmogorov hypotheses, with new perturbation and closure methods.

Contribution

It introduces a novel multiscale wavelet approach to turbulence, incorporating stochastic formalism and energy transfer analysis, advancing theoretical understanding.

Findings

Loop divergences are canceled in the wavelet-based stochastic expansion.

The energy transfer from large to small scales is explicitly modeled.

Kolmogorov hypotheses are naturally reformulated in multiscale formalism.

Abstract

We present a multiscale description of hydrodynamic turbulence in incompressible fluid based on a continuous wavelet transform (CWT) and a stochastic hydrodynamics formalism. Defining the stirring random force by the correlation function of its wavelet components, we achieve the cancellation of loop divergences in the stochastic perturbation expansion. An extra contribution to the energy transfer from large to smaller scales is considered. It is shown that the Kolmogorov hypotheses are naturally reformulated in multiscale formalism. The multiscale perturbation theory and statistical closures based on the wavelet decomposition are constructed.