Analysis of Turburence by a Statistics based on Non-Extensive Entropy

An analysis of fully developed turbulence is developed based on the assumption that the underlying statistics of the system is of Tsallis ensemble. The multifractal spectrum $f_{\mathrm{T}}(\alpha) $ corresponding to the Tsallis type distribution function is deter-mined self-consistently in the sense that all parameters can be obtained through the observed value of the intermittency exponent. It is shown that the scaling exponents $(_{m} $ of velocity structure function derived with the help of the multifractal spectrum fits very well with experimental data. It is revealed that the asymptotic expression of $\zeta_{m} $ for $m>>1 $ has a $\log $ term, which was not recognized before. The present self-consistent approach narrowed down the value of intermittency exponent $\mu $ for the fully developed turbulence to $\mu=0.235\pm 0.015 $. In a previous paper [1], we showed that Tsallis index $q[2,3] $ corresponding to the $\mathrm{p}-$ model [4] can be effectively determined by observed values of the intermittency exponent $\mu $ with the help of the scaling relation (16) below [5]. We proposed in [1] a Tsallis type distribution function for the probability density function of the local dissipation, an