Analysis of velocity fluctuation in turbulence based on generalized statistics

Abstract

The numerical experiments of turbulence conducted by Gotoh et al. are analyzed precisely with the help of the formulae for the scaling exponents of velocity structure function and for the probability density function of velocity fluctuations. These formulae are derived by the present authors with the multifractal aspect based on the statistics that are constructed on the generalized measures of entropy, i.e., the extensive Rényi's or the non-extensive Tsallis' entropy. It is shown, explicitly, that there exist two scaling regions, i.e., the upper scaling region with larger separations which may correspond to the scaling range observed by Gotoh et al., and the lower scaling region with smaller separations which is a new scaling region extracted first by the present systematic analysis. These scaling regions are divided by a definite length approximately of the order of the Taylor microscale, which may correspond to the crossover length proposed by Gotoh et al. as the low end of the scaling range (i.e., the upper scaling region). 47.27.-i, 47.53.+n, 47.52.+j, 05.90.+m After the discovery of the Kolmogorov spectrum [1], there have been plenty of trials including [2,3] in order to understand the intermittent evolution of fluid in fully developed turbulence. Among them, the line based on a kind of ensemble theoretical approaches, initiated by the log-normal model [4][5][6], consists of the β-model (a unifractal dimensional analysis) [7], the p-model (a multifractal model) [8,9], the 3D binomial Cantor set model [10] and so on. On this line, an investigation of turbulence based on the generalized entropy, i.e., Rényi's [11] or Tsallis' [12,13], was started by the present authors [14][15][16][17][18][19][20]. After a rather preliminary investigation of the p-model [14], it has been developed further to derive the analytical expression for the scaling exponents of velocity structure function [15][16][17][18], and to determine the probability density function (PDF) of velocity fluctuations [18][19][20] by a self-consistent statistical mechanical approach.With the help of the analytical formulae derived in [15][16][17][18][19][20], we will analyze, in this paper, the PDF's of velocity fluctuations observed in the beautiful DNS (i.e., the direct numerical simulation) conducted by Gotoh et al. [21]. We will deal with the data at the Taylor microscale Reynolds number R λ = 381, since at this Reynolds number Gotoh et al. observed the PDF with accuracy up to order of 10 −9 ∼ 10 −10 , in contrast with any previous experiments, real or numerical. We showed in [19] that our formulae can explain quite well the PDF's observed in the real experiment by Lewis and Swinney [22] for turbulent Couette-Taylor flow at R λ = 270 (Re = 5.4 × 10 5 ) produced in a concentric cylinder system in which, however, the PDF's were measured only with accuracy of order of 10 −5 . Note that the success of the present theory in the analysis of this turbulent Couette-Taylor flow may indicate the robustness of singularities associated with velocity grad...

Analysis of Accelerations in Turbulence Based on Generalized Statistics

“…The formula quite well explains the experimental data [2][3][4][5][6]8]. Note that the formula is independent of the length n , and, therefore, independent of n. Let us derive the PDFΛ (n) (ω n ) of the accelerations by dividing it into two parts with respect to ω n , i.e.,…”

“…We expect that other data for the same system, in addition to the PDF of accelerations, will be provided in the near future, such as the intermittency exponent µ, the scaling exponents ζ m , the PDF's of the velocity fluctuations or of the velocity derivatives. Then, we can cross-check the validity of the present analysis based on the formula (8). 4 Note that the formula for the PDF of accelerations is different from the one for the PDF's of velocity fluctuations or of velocity derivatives.…”

“…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.…”

“…For the region where the value of µ is usually observed, i.e., 0.13 µ 0.40, the three self-consistent equations are solved to give the approximate equations [8]: α 0 = 0.9989 + 0.5814µ, X = −2.848 · 10 −3 + 1.198µ and q = −1.507 + 20.58µ − 97.11µ 2 + 260.4µ 3 − 365.4µ 4 + 208.3µ 5 . Since there are two mechanisms in a turbulent flow to rule its evolution, i.e., the one controlled by the kinematic viscosity that takes care of thermal fluctuations, and the other by the turbulent viscosity that is responsible for intermittent fluctuations related to the singularities in acceleration, it may be reasonable to assume that the probability Λ (n) (y n )dy n to find the scaled pressure fluctuations |y n | = δp n /δp 0 in the range y n ∼ y n + dy n has two independent origins:…”

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 ).…”