A parsimonious and universal description of turbulent velocity increments

Abstract

We discuss an analysis of the probability density function (pdf) of turbulent velocity increments based on the class of normal inverse Gaussian distributions. It allows for a parsimonious description of velocity increments that covers the whole range of amplitudes and all accessible scales from the finest resolution up to the integral scale. The analysis is performed for three different data sets obtained from a wind tunnel experiment, a free-jet experiment and an atmospheric boundary layer experiment with Taylor-Reynolds numbers R λ = 80, 190, 17000, respectively. The application of a time change in terms of the scale parameter δ of the normal inverse Gaussian distribution reveals some universal features that are inherent to the pdf of all three data sets.

The Kolmogorov–Obukhov Statistical Theory of Turbulence

“…It has an algebraic cusp at the origin and exponential tails and is constructed by multiplying the modified Bessel's function of the second kind K λ , by x −λ . For the zeroth moment we get a distinguished solution λ =c = 1 which give the Normal Inverse Gaussian (NIG) distribution that was also investigated by BarndorffNilsen [5] and used by Barndorff-Nilsen, Blaesild and Schmiegel to model PDF of velocity increments for several data sets in [7]. It turns out that the distribution functions for all of the moments can be expressed by the NIG distribution function.…”

“…Because of intermittency each structure function generates its own NIG distribution with separate parameters. Then we compare these PDFs with DNS results and experimental data, see also [6,7].…”

“…The NIG distribution has very nice properties that are summarized in [7]. In particular its characteristic function and all of its moments are easily computed.…”

“…is particularly simple for the NIG and this make the moments easy to compute, see [7]. The first few moments and the characteristic function of the NIG distribution are:…”

Infinite Divisibility for Stochastic Processes and Time Change

“…This question is considered for Brownian subordination in Geman, Madan and Yor (2002) and their work has been extended by Winkel (2001) to time change of Brownian motion with more general chronometers. For some discussions of time change in quantum physics and in turbulence see Chung and Zambrini (2003) and Barndorff-Nielsen, Blaesild and Schmiegel (2004), respectively. Time change in a broad mathematical sense is treated in Barndorff-Nielsen and Shiryaev (2005).…”