Analysis of velocity fluctuation inturbulence based on generalized statistics

Abstract

The numerical turbulence experiments conducted by Gotoh et al areanalysed with high precision with the help of formulae for thescaling exponents of the velocity structure function and for theprobability density function of the velocity fluctuations. Theseformulae are derived by the present authors, with the multifractalaspect based on statistics constructed using generalized measures ofentropy, i.e., the extensive Rényi entropy or the non-extensiveTsallis entropy. It is shown, explicitly, that there exist two scaling regions, i.e., the upper scaling region withlarger separations which may correspond to the scaling rangeobserved by Gotoh et al and the lower scaling region withsmaller separations which is a new scaling region, extracted for thefirst time by the present systematic analysis. These scaling regions aredivided by a definite length approximately of the order of theTaylor microscale, which may correspond to the crossover lengthproposed by Gotoh et al as the low end of the scaling range (i.e.,the upper scaling region).