Analysis of Velocity Derivatives in Turbulence based on Generalized Statistics

– A theoretical formula for the probability density function (PDF) of velocity derivatives in a fully developed turbulent flow is derived with the multifractal aspect based on the generalized measures of entropy, i.e., the extensive Rényi entropy or the non-extensive Tsallis entropy, and is used, successfully, to analyze the PDF’s observed in the direct numerical simulation (DNS) conducted by Gotoh et al.. The minimum length scale rd/η in the longitudinal (transverse) inertial range of the DNS is estimated to be r d/η = 1.716 (r T d /η = 2.180) in the unit of the Kolmogorov scale η. In the previous paper [1], we analyzed, precisely, the probability density functions (PDF’s) of velocity fluctuations observed in the direct numerical simulation (DNS) of turbulence conducted by Gotoh et al. [2] at the Taylor microscale Reynolds number Rλ = 381. At this Reynolds number, the PDF’s had been measured with high accuracy up to the order of 10 ∼ 10. The correctness of the analytical formula for the scaling exponents of velocity structure function [3–6] enabled us to extract the value of the intermittency exponent μ by fitting it with the ten observed data in the DNS by the method of least square [1]. With the intermittency exponent, the parameters in the analytical formula of the PDF of velocity fluctuations [6–8] are determined, self-consistently. The formula was used to fit each of the observed PDF’s of velocity fluctuations for ten different separations r/η by means of the method of least square [1], and to extract, successfully, the dependence of the number n of steps in the energy cascade on the separation r/η: n = −1.050× log2 r/η + 16.74 (for l L c ≤ r), (1) n = −2.540× log2 r/η + 25.08 (for r < l L c ) (2) with the crossover length lc /η = 48.26 for longitudinal fluctuations, and n = −0.9896× log2 r/η + 13.95 (for l T c ≤ r), (3) n = −2.820× log2 r/η + 23.87 (for r < l T c ) (4)