Tsallis statistics and turbulence☆

Analysis of velocity fluctuation in turbulence based on generalized statistics

“…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.…”

“…is the multifractal spectrum [15][16][17][18], derived by the relation P (n) (α) ∝ δ 1−f (α) n [9,18], that reveals how densely each singularity, labeled by α, fills physical space.…”

“…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.…”

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

“…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 revealed also that, within the same line of the statistical mechanical treatment, the PDF of the velocity fluctuations can be derived analytically [12,13].…”

“…The parameters appearing in the ensemble theory are determined self-consistently with the help of the physically meaningful equations, i.e., the conservation of energy, the definition of the intermittency exponent and the scaling relation relevant to the analysis by statistics based on the generalized entropy [9][10][11][12][13]. The present ensemble theoretical method rests on the following aspects: 1) the fully developed turbulence is the situation where the kinematic viscosity can be neglected compared with the effect of the turbulent viscosity that is a cause of intermittency; 2) in this situation, one can find an invariance of the Navier-Stokes equation in certain scale transformation which contains an arbitrary exponent that measures the degrees of singularity in velocity gradient; 3) for the distribution of the exponent, the statistics based on the generalized entropies is adopted, which was shown [10][11][12][13] to be a generalization of the statistics in the log-normal model [14][15][16]. The statistical mechanical procedure based on the Boltzmann-Gibbs entropy gives the results of the log-normal model, whereas the one based on the generalized entropies provides us with the above mentioned new results [12].…”