The renormalization group, the ?-expansion and derivation of turbulence models

Abstract

We reformulate the renormalization group (RNG) and theɛ-expansion for derivation of turbulence models. The procedure is developed for the Navier-Stokes equations and the transport equations for the kinetic energyK and energy dissipation rate ℰ. The derivation draws on the works of Yakhot and Orszag (1986) and Smith and Reynolds (1992), and all results are found at low order in the underlying perturbation expansion in powers ofɛ. The sum of the source terms in the ℰ-equation is known to beO(1) due to the balance at leading order ofO(R T 1/2 ) terms. Smith and Reynolds (1992) showed the cancellation of some of theO(R T 1/2 ) terms generated by the RNG procedure. Here we show that including the random-force contribution to ℰ-production results in the cancellation ofall theO(R 1/2 ) terms. We find that two of theO(1) terms in the RNG equation for the mean dissipation rate ℰ have the same form as those in the widely used model -equation. The values of the coefficients of the familiar terms are close to those used in practice. An extra production term is predicted which is small for slow distortions, but important for rapid distortions. Hence, it may be a term that should be added to the model equation. We believe that the present derivation places the model equation on a more solid theoretical basis.