Analysis of velocity fluctuation in turbulence based on generalized statistics

The numerical turbulence experiments conducted by Gotoh et al are analysed with high precision with the help of formulae for the scaling exponents of the velocity structure function and for the probability density function of the velocity fluctuations. These formulae are derived by the present authors, with the multifractal aspect based on statistics constructed using generalized measures of entropy, i.e., the extensive Renyl entropy 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 for the first time 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).