Analysis of Fully Developed Turbulence by a Generalized Statistics

Abstract

In order to reveal the underlying statistics properly describing fully developed turbulence, the probability density function of the local dissipation is determined by taking the extreme of a generalized entropy (Tsallis entropy). With the density function, the scaling exponents ζ m of the velocity structure function are derived analytically. It is found that these scaling exponents are consistent with experimental data. The asymptotic expression of ζ m for m 1 has a log term.In a previous paper, 1) we showed that the Tsallis index q 2), 3) corresponding to the p-model 4) can be effectively determined by observed values of the intermittency exponent µ with the help of the scaling relation 5)where α max and α min represent the extremes of the multifractal spectrum f (α) within the p-model. We proposed in Ref. 1) a Tsallis-type distribution function for the probability density function of the local dissipation, and showed that the proposed distribution function with the derived Tsallis index q explains the binomial distribution function of the p-model very well.In this paper, we develop the program presented in Ref. 1) in order to reveal the underlying statistics properly describing fully developed turbulence. 6), 7) We seek an appropriate probability density function of the local dissipation by taking the extreme of a generalized entropy (called the Tsallis entropy 2), 3) ) under two constraints: normalization of the probability and fixing the intermittency exponent µ to a constant value. The latter constraint is a manifestation of the situation that the only information we have is the intermittency exponent µ. The Tsallis entropy includes the Boltzmann-Gibbs entropy as a special case. We determine the multifractal spectrum f (α) corresponding to the probability density function self-consistently in the sense that all quantities can be calculated by using the observed value of the intermittency exponent µ. There is no fitting parameter left. The probability density function we obtain reveals that the underlying statistics are not of the extensive Boltzmann-Gibbs-type but of the non-extensive Tsallis-type.With the help of the multifractal spectrum f (α), we derive the scaling exponents ζ m of the velocity structure function [see (5) below], and compare them with experimental data and with the curves given by the K41, log-normal, β-model, pmodel and log-Poisson theories. We show that the present result accounts for all the by guest on June 4, 2015 http://ptp.oxfordjournals.org/ Downloaded from

Analysis of velocity fluctuation in turbulence based on generalized statistics

“…We have already made clear in [20] that the present theory can also explain quite well the PDF's of longitudinal velocity fluctuations reported by Gotoh et al [21], and have revealed the superiority of our PDF to the one derived in [23] when the accuracy is raised up to the order of 10 −9 . Assured by the success of these rather preliminary tests, we will apply our theory for further precise analyses of the data obtained in [21] including the PDF's of transverse velocity fluctuations in addition to those of longitudinal fluctuations.…”

“…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 gradient even for the case of no inertial range.…”

“…These equations are slightly different from those given in [19], since the region of µ has been extended a little bit. If we put q = 1 from the beginning, i.e., starting with the Boltzmann-Shannon entropy, and take an extremum with the same constraints given above for q = 1, we have a Gaussian distribution function.…”

“…is the so-called scaling exponent of the velocity structure function, whose expression was derived first by the present authors [15][16][17][18]. In this paper, we will use the formula (6) in order to extract the value of the intermittency exponent µ for the best fit to the measured scaling exponents by the method of least squares.…”

“…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.…”