Statistical laws and vortex structures in fully developed turbulence

Abstract

Fully developed turbulence is structured with a number of intense elongated vortices. Recognizing that statistical laws are related to such structures, the flow field is modeled by an ensemble of strained vortices (i.e. Burgers vortices) distributing randomly in space, from which probability density functions (pdfs) for longitudinal and transversal components of velocity difference are derived by taking statistical averages ensuring isotropy and homogeneity for the velocity field. It is found 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, with a transition scale between the two ranges being at the order of mean size of Burgers vortices. Velocity structure functions show scaling behaviors in the second interval corresponding to the inertial range with the scaling exponents close to those known in the experiments and direct numerical simulations. 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 a recent experiment. 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.