Renormalization-group approach to the stochastic Navier--Stokes equation: Two-loop approximation

Abstract

The field theoretic renormalization group is applied to the stochastic Navier--Stokes equation that describes fully developed fluid turbulence. The complete two-loop calculation of the renormalization constant, the $β$ function, the fixed point and the ultraviolet correction exponent is performed. The Kolmogorov constant and the inertial-range skewness factor, derived to second order of the $\eps$ expansion, are in a good agreement with the experiment. The possibility of the extrapolation of the $\eps$ expansion beyond the threshold where the sweeping effects become important is demonstrated on the example of a Galilean-invariant quantity, the equal-time pair correlation function of the velocity field. The extension to the $d$-dimensional case is briefly discussed.