Velocity difference statistics in turbulence

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 co-workers, 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, Physica D 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, Physica A 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.