Analysis of Accelerations in Turbulence Based on Generalized Statistics

Abstract

An analytical expression of probability density function (PDF) of accelerations in turbulence is derived with the help of the statistics based on generalized entropy (the Tsallis entropy or the Rényi entropy). It is revealed that the derived PDF explains the one obtained by Bodenschatz et al. in the measurement of fluid particle accelerations in fully developed turbulence at R λ = 970 . 47.27.-i, 47.53.+n, 47.52.+j, 05.90.+m The multifractal analysis of turbulence by the statistics based on the generalized entropy of Rényi's or of Tsallis' has been developed by the present authors [1][2][3][4][5][6][7][8][9]. Similarly to the usual thermodynamic entropy, the Rényi entropy [10] is of an extensive character, whereas the Tsallis entropy [11-13] is non-extensive. The multifractal analysis belongs to the line of study based on a kind of ensemble theoretical approaches that, starting from the log-normal model [14][15][16], continues with the β-model [17], the p-model [18,19], the 3D binomial Cantor set model [20] and so on. After a rather preliminary investigation of the p-model [1], we developed further to derive the analytical expression for the scaling exponents of velocity structure function [2][3][4][5], and to determine the probability density function (PDF) of velocity fluctuations [5][6][7][8] and of velocity derivative [9] by a self-consistent statistical mechanical approach. Key words: multifractal analysis, fully developed turbulence, PDF of fluid particle accelerations, Rényi entropy, Tsallis entropy PACS:In this paper, we will derive the formula for the PDF of the accelerations of a fluid particle in fully developed turbulence by means of the multifractal analysis. With the theoretical PDF, we will analyze the PDF of accelerations at R λ = 970 (the Taylor microscale Reynolds number) obtained in the Lagrangian measurement *

Harmonious representation of PDFs reflecting large deviations

“…For example, through the analysis of the velocity fluctuations in Fig. 1, we extracted the dependence of q ′ on r/η as [9] q ′ = 1.71 − 0.05 log 2 (r/η). We then obtain the diffusion coefficient for the fluctuations of magnitude |ξ n | (≤ ξ * n ) in the form D φ (ξ n )/D φ (0) = 1 + [0.71 − 0.05 log 2 (r/η)][(φ + 3f ′ (α * ))/2φ](ξ n /ξ * n ) 2 .…”

“…For values of the variable |ξ n | larger than the order of its standard deviation, ξ * n ≤ |ξ n | ≤ ξ max n (equivalently, α min ≤ |α| ≤ α * ), the PDF is given by [8,9] …”

“…It has been proposed by the present authors [1,2,3,4,5,6,7,8,9] in order to analyze various probability density functions (PDF's) observed in the system of fully developed turbulence. The PDF's, given for variables normalized by their own standard deviations, represent a characteristic large wings (or fat tails) due to the intermittency specific to turbulence.…”

“…The MFA is constructed with the help of the Tsallis-type distribution function [16] that provides an extremum of the extensive Rényi entropy [17] or the non-extensive Tsallis entropy [16,18] under appropriate constraints [3]. After a rather preliminary investigation of the p model [1], we developed further to derive the analytical expression for the scaling exponents of velocity structure function [2,3], and to determine the PDF's of the velocity fluctuations [3,4,5], of the velocity derivative [6] and of the fluid particle accelerations [7].…”

“…In order to perform a transparent comparison, the framework of MFA is rephrased in its most sophisticated fashion in which the tail part of PDF, giving the probabilities of events larger than its standard deviation, is written down once the multifractal spectrum for the spatial distribution of singularities is specified (see (1) below), whereas the center part, giving the probabilities of events smaller than its standard deviation, is assumed to be analyzed by the Tsallis-type PDF for the variable itself [8,9] (see (2) below). The various PDF's extracted out from the DNS conducted by Gotoh et al [19,20], and the PDF of fluid particle accelerations observed by Bodenschatz et al in their Lagrangian measurement of particle accelerations [21,22,23] will be analyzed with the help of the multifractal spectrums for the log-normal, the p model and the harmonious representation.…”