Conditional mode elimination and scale-invariant dissipation in isotropic turbulence

We show that a conditional average (based on a limit in which the Fourier m odes of the turbulent velocity field with wavenumber k less than or equal t o k(C), where k(C) is an arbitrary cutoff wavenumber, are held constant) ca n be used to separate the nonlinear coupling to high-k modes into coherent and random parts, with the latter rigorously determining the net energy tra nsfer. In addition, we show that three symmetry-breaking terms, which are g enerated by the conditional average of the Navier-Stokes equation, do not c ontribute to the equation for the energy dissipation. Two of these terms va nish identically, under unconditional averaging and wavenumber integration, respectively, and the remaining one vanishes in the limit of asymptotic fr eedom (when calculated by a quasi-stochastic estimate, from the high-k mome ntum equation). If the cutoff k(C) is chosen to be large enough, then the c onditionally averaged high-k equation can be solved perturbatively in terms of the local Reynolds number which is less than unity. On this basis, an e xpression for the renormalized dissipation rate is obtained as an expansion in a parameter (lambda) which is equal to the square of the local Reynolds number. A recursive calculation is made of the renormalized dissipation ra te, in which the expansion parameter reaches a maximum value of lambda = 0. 16 at the fixed point. It is also shown that a previous Markovian approxima tion can be replaced by an exact summation to consistent order in perturbat ion theory. (C) 2001 Elsevier Science B.V. All rights reserved.