Field-theoretic renormalization group and turbulence

The recent analysis of the fluctuating Navier-Stokes equations with a random force with a $k^{\mathrm{-}y}$ spectrum by Yakhot and Orszag [Phys. Rev. Lett. 57, 1722 (1986)] is repeated using field-theoretical renormalization techniques. A dimensional expansion around the full equilibrium state in two spatial dimensions is performed to one-loop order. The results differ from those of Yakhot and Orszag in a number of ways. A nontrivial fixed point is found which is the natural extension of that presented by Forster, Stephen, and Nelson [Phys. Rev. A 16, 732 (1977)], and various correlation functions in crossover form are presented. The application of the model to turbulence, i.e., in cases where the Kolmogorov (5/3 law is obtained, is analyzed in detail. Unlike the work of Yakhot and Orszag, a choice of the random force correlation exponent (y=-1.5851 in three dimensions) is found which gives the Kolmorov (5/3 law at high wave vectors.