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 in d > 2 dimensions. For the first time, 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 are expressed in terms of universal (in the sense of the theory of critical behavior) quantities, which allows one to construct for them a regular perturbative calculational scheme. The practical calculations are performed up to the second order of the ε-expansion (two-loop approximation). The results obtained are in a good agreement with the experiment. The large d behavior of these quantities is briefly discussed. The possibility of the extrapolation of the ε-expansion beyond the threshold where the sweeping effects become important is demonstrated by the example of a Galilean-invariant quantity, the equal-time pair correlation function of the velocity field.