Statistical Laws Governed by Vortex Structures in Fully Developed Turbulence

Abstract

Statistics of fully developed turbulence is modeled by an ensemble of strained vortices (i.e. Burgers vortices) distributing randomly in space, and probability density functions (pdfs) for longitudinal and transversal components of velocity difference are estimated by taking statistical averages of isotropy and homogeneity. It is found [15] that the pdfs tend to close-to-exponential forms at small scales, and that there exist two scaling ranges in the structure function of every order, which are identified as the viscous range and inertial range respectively. The p dfs deviate increasingly away from the Gaussian as the separation distance decreases. For the inertial range (second scaling range of larger scales), scaling exponents are obtained and found to be close to those known in the experiments and DNSs. It is remarkable that the Kolmogorov’s four-fifths law is observed to be valid in a small-scale range. The scaling exponents of higher order structure functions are numerically estimated up to the 25th order. It is found that asymptotic scaling exponents as the order increases are in good agreement with the behavior of recent experiment of Praskovsky & Oncley [4]. The above model analysis is considered to represent successfully the statistical behaviors at small scales (possibly less than the Taylor microscale) and higher orders. The present statistical analysis leads to scale-dependent probability density functions.