Abstract

We have used wavelets to analyse, model and compute turbulent flows. The theory and open questions encountered in turbulence are presented. The wavelet-based techniques that we have developed to study turbulence are explained and the main results are summarized.

Introduction

In this chapter we will summarize the ten years of research we have done to try to better understand, model and compute fully developed turbulent flows using wavelets and wavelet packets. Fully developed turbulence is a highly nonlinear regime (very large Reynolds number tending to infinity) and is distinct from the transition to turbulence (low Reynolds number). We have chosen to present a personal point of view concerning the current state of our understanding of fully developed turbulence. It may not always coincide with the point of view of other researchers in this field because many issues we are addressing in this chapter are still undecided and highly controversial. This paper is a substantially revised and extended version of: Wavelets and Turbulence by Farge, Kevlahan, Perrier and Goirand which appeared in Proceedings of the IEEE, vol. 84, no. 4, April 1996, pp. 639–669.

After more than a century of turbulence study [30], [173], no convincing theoretical explanation has produced a consensus among physicists (for a historical review of various theories of turbulence see [160], [158], [72], [91]). In fact, a large number of ad hoc ‘phenomenological’ models exist that are widely used by fluid mechanicians to interpret experiments and to compute many industrial applications (in aeronautics, combustion, meteorology …) where turbulence plays a role.