Renormalization group in the infinite-dimensional turbulence: third-order results

TL;DR

This paper applies the field theoretic renormalization group to the stochastic Navier-Stokes equation in high dimensions, deriving third-order results for turbulence characteristics and showing promising agreement with experiments.

Contribution

It develops a three-loop epsilon expansion for turbulence in the large-d limit, simplifying diagram calculations and providing new quantitative predictions.

Findings

Calculated renormalization constants, beta function, and correction exponent to order epsilon^3.

Derived third-order expressions for the Kolmogorov constant and inertial-range skewness.

Results for the Kolmogorov constant agree reasonably with experimental data.

Abstract

The field theoretic renormalization group is applied to the stochastic Navier-Stokes equation with the stirring force correlator of the form k^(4-d-2\epsilon) in the d-dimensional space, in connection with the problem of construction of the 1/d expansion for the fully developed fluid turbulence beyond the scope of the standard epsilon expansion. It is shown that in the large-d limit the number of the Feynman diagrams for the Green function (linear response function) decreases drastically, and the technique of their analytical calculation is developed. The main ingredients of the renormalization group approach -- the renormalization constant, beta function and the ultraviolet correction exponent omega, are calculated to order epsilon^3 (three-loop approximation). The two-point velocity-velocity correlation function, the Kolmogorov constant C_K in the spectrum of turbulent energy and the inertial-range skewness factor S are calculated in the large-d limit to third order of the epsilon expansion. Surprisingly enough, our results for C_K are in a reasonable agreement with the existing experimental estimates.