Velocity difference statistics in turbulence

TL;DR

This paper unifies two approaches explaining non-Gaussian velocity difference PDFs in turbulence, applies the combined method to Couette-Taylor data, and determines subsystem size, revealing a log-normal distribution of the intensive parameter.

Contribution

It unifies Castaing and Beck-Cohen methods for turbulence velocity PDFs and introduces a way to determine subsystem size in the Beck-Cohen approach.

Findings

The combined approach fits experimental turbulence data well.

Subsystem size can be determined within the Beck-Cohen framework.

The intensive parameter follows a log-normal distribution.

Abstract

We unify two approaches that have been taken to explain the non-Gaussian probability distribution functions (PDFs) obtained in measurements of longitudinal velocity differences in turbulence, and we apply our approach to Couette-Taylor turbulence data. The first approach we consider was developed by Castaing and coworkers, who obtained the non-Gaussian velocity difference PDF from a superposition of Gaussian distributions for subsystems that have a particular energy dissipation rate at a fixed length scale [Castaing et al., {\it Physica D} {\bf 46}, 177 (1990)]. Another approach was proposed by Beck and Cohen, who showed that the observed PDFs can be obtained from a superposition of Gaussian velocity difference PDFs in subsystems conditioned on the value of an intensive variable (inverse ``effective temperature'') in each subsystem [Beck and Cohen, {\it Physica A} {\bf 322}, 267 (2003)]. The intensive variable was defined for subsystems assuming local thermodynamic equilibrium, but no method was proposed for determining the size of a subsystem. We show that the Castaing and Beck-Cohen methods are related, and we present a way to determine subsystem size in the Beck-Cohen method. The application of our approach to Couette-Taylor turbulence (Reynolds number $540~000$) yields a log-normal distribution of the intensive parameter, and the resultant velocity difference PDF agrees well the observed non-Gaussian velocity difference PDFs.