Analysis of turbulence by statistics based on generalized entropies

An analytical formula of the scaling exponents of velocity structure function for fully developed turbulence is derived, non-perturbatively, by assuming that its underlying statistics is the one based on the generalized measures of entropy, the Renyi entropy or the Havrda–Charvat–Tsallis (HCT) entropy. It is revealed by a self-consistent analysis for the observed value μ=0.220 (±1%) that the formula explains experimental data very well with single value, q=0.343, of the index which appears in the measures of the Renyi entropy or of the HCT entropy. The probability density functions of the velocity fluctuation and of the velocity gradient are also presented.