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

Abstract

The field theoretic renormalization group is applied to the stochastic Navier–Stokes equation with the stirring force correlator of the form k4−d−2ε 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 ε 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, the β function and the ultraviolet correction exponent ω, are calculated to order ε3 (three-loop approximation). The two-point velocity–velocity correlation function, the Kolmogorov constant CK in the spectrum of turbulent energy and the inertial-range skewness factor are calculated in the large-d limit to the third order of the ε expansion. Surprisingly enough, our results for CK are in reasonable agreement with the existing experimental estimates.