Multi-Scale Analysis of Turbulence

The basic idea of the processes taking place in fully developed turbulence is still that of a cascade in which vortices break up successively into smaller ones. A result of this directed process is the emergence of intermittency, i.e. extraordinary strong fluctuations on small scales. An objective is to describe and to understand the statistics of the velocity fluctuations on differerent scales. Based on the pioneer work of Kolmogorov the cascade resembles a process being self-similar on different scales but with deviations due to intermittency. But this description, which is based on scaling property of structure functions, is too idealized for moderate Reynolds numbers and is not complete as discussed below.

It has been shown by Friedrich and Peinke [1] that the measured velocityfluctuations on different scales can be well described by a diffusion process. Experimentally almost no deviation from this description can be found for a variety of flows and for a large range of scales [2]. Moreover this description is complete in the sense that it grasps also all statistical relations

between

the different scales. The result is a Fokker-Planck equation, which can be estimated in a parameter-free way directly from data and which can be used for subsequent analytical considerations.

In this paper we give an overview of some central results using this analysis. We deal in essential with four different points:

At scales below the Taylor-scale, the here proposed statistical description of the cascade is not possible anymore. We discuss this point and relate the limiting scale to a new Einstein-Markov-coherence length.

We show how this diffusion process in scale can be used to generate time series with identical scale dependent statistics.

We discuss the interactions between different velocity components in terms of a multi-variate Fokker-Planck equation and explain their debated differences.

We discuss the Reynolds number dependence of a turbulent field and discuss the asymptotic state for infinite high Reynolds numbers.