PDF of velocity fluctuation in turbulence by a statistics based on generalized entropy

Abstract

An analytical formula for the probability density function (PDF) of the velocity fluctuation in fully-developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the Rényi entropy or the Tsallis entropy. The parameters appearing in the PDF, including the index q in the generalized measures, are determined self-consistently with the help of the observed value µ of the intermittency exponent. The derived PDF explains quite well the experimentally observed density functions. In this paper, we analyze the probability density function (PDF) of the velocity fluctuations in turbulence measured in the experiments conducted by Lewis and Swinney [1] and by Gotoh et al. [2] in terms of a generalized statistics constructed on Rényi's [3] or on Tsallis' entropy [4,5]. The Rényi entropy has an information theoretical basis, and has the extensive characteristics as the usual thermodynamical entropy does. On the other hand, the Tsallis entropy is non-extensive, and therefore it provides us with one of the attractive trials for generalized statistical mechanics which deals with, for example, systems with long range correlations. Note that the distribution functions which give an extremum of each entropy have a common structure (see (5) for the present system) in spite of different characteristics of these entropies. The investigation of turbulence based on the generalized statistics was set about by the present authors [6] with an investigation of the p-model [7,8]. Then, the treatment was sublimated by a self-consistent analysis following the usual procedure of ensembles in statistical mechanics, where it was shown that the derived analytical expression for the scaling exponents of the velocity structure function explains quite well the experimentally observed exponents [9][10][11][12]. It has been

Analysis of Accelerations in Turbulence Based on Generalized Statistics

“…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][12][13] is non-extensive.…”

Multifractal Analysis of Various PDF in Turbulence based on Generalized Statistics: A Way to Tangle in Superfluid He

“…Through the analyses of the PDFs for velocity fluctuations in Fig. 2, we extracted the formula for the dependence of n on r/η: [7,8] n = −0.989 × log 2 r/η + 16.1 (for ℓ c ≤ r), (8) n = −2.40 × log 2 r/η + 24.0 (for r < ℓ c ).…”

“…In this paper, we derive the unified formula for various probability density functions (PDFs) in fully developed turbulence by means of the multifractal analysis (MFA) [1,2,3,4,5,6,7,8,9,10,11,12], and analyze the PDFs observed in two experiments, i.e., the PDFs of velocity fluctuations, of velocity derivatives and of fluid particle accelerations at R λ = 380 that was extracted by Gotoh et al from the DNS of the size 1024 3 [13], and the PDF of fluid particle accelerations at R λ = 690 obtained in the Lagrangian measurement of particle accelerations that was realized by Bodenschatz and co-workers [14,15,16] by raising dramatically the spatial and temporal measurement resolutions with the help of the silicon strip detectors. The MFA of turbulence is a unified self-consistent approach for the systems with large deviations, which has been constructed based on the Tsallistype distribution function [17,18] that provides an extremum of the extensive Rény [19] or the non-extensive Tsallis entropy [17,18,20] under appropriate constraints.…”

Nonextensive statistical mechanics: a brief review of its present status

“…As important as this is the fact that both Bediaga et al and Beck theories recover a crucial feature of Hagedorn's scenario, namely that the temperature to be associated with the distributions should not depend on the collision energy. [18] for details). The entire set of theoretical curves has been obtained with a single value q sen < 1.…”