Equifinality and the Scaling Exponent of the Structure Function

Abstract:

In turbulence the structure function is by far the most widely used tool for the empirical analysis of the velocity field. This is due mainly to the work of Kolmogorov (1941) who hypothesised a homogeneous flux of energy and derived the famous 2/3 power law for the second-order structure function; — which corresponds to a 5/3 law for the energy spectrum (Obukhov, 1942).

In 1962 Kolmogorov refined his hypothesis to take into account the intermittency of the flux, with the consequence that the exponent ξ(q) of the structure function is not longer proportional to its statistical order q.

In this communication, we first show that the refined hypothesis can lead to different models that can have opposite intermittency corrections. Secondly, we demonstrate that the inverse problem, i.e., starting from a given expression of ξ(q) to recover the involved flux leads to an interesting problem of equifinality for the definition of this flux. This is done in particular in the framework of the Fractionally Integrated Flux model that gives a precise meaning to the refined hypothesis. The theoretical and practical consequences are illustrated with the help data analysis and simulations of turbulence in wind farms and urban lakes.