Cross-term and ε-expansion in RNG theory of turbulence

The renormalization group theory of turbulence by Yakhot and Orszag (J. Sci. Comput. 1 (1986) 3) has produced many interesting and useful results yet some of its foundational aspects require clarification. One of such aspects is the use of the expansion parameter ε in the expressions for renormalized characteristics of fully developed turbulence. In the one-loop approximation of the original Yakhot-Orszag theory, the value of ε=0 is enforced in the amplitude of the renormalized viscosity via fixed point arguments whereas it is set at 4 elsewhere, as required by the exponent of the power spectrum. Another problematic issue is the treatment of the term representing triple nonlinearity. Here we show that many of these issues can be resolved with proper accounting for the cross-term, or the product of fast and slow modes. The resulting model is self-consistent and preserves all the original results of the Yakhot-Orszag theory.