Conditional mode elimination and scale-invariant dissipation in isotropic turbulence

Abstract

We show that a conditional average (based on a limit in which the Fourier modes of the turbulent velocity field with wavenumber k⩽ k C , where k C is an arbitrary cutoff wavenumber, are held constant) can be used to separate the nonlinear coupling to high- k modes into coherent and random parts, with the latter rigorously determining the net energy transfer. In addition, we show that three symmetry-breaking terms, which are generated by the conditional average of the Navier–Stokes equation, do not contribute to the equation for the energy dissipation. Two of these terms vanish identically, under unconditional averaging and wavenumber integration, respectively, and the remaining one vanishes in the limit of asymptotic freedom (when calculated by a quasi-stochastic estimate, from the high- k momentum equation). If the cutoff k C is chosen to be large enough, then the conditionally averaged high- k equation can be solved perturbatively in terms of the local Reynolds number which is less than unity. On this basis, an expression for the renormalized dissipation rate is obtained as an expansion in a parameter ( λ) which is equal to the square of the local Reynolds number. A recursive calculation is made of the renormalized dissipation rate, in which the expansion parameter reaches a maximum value of λ=0.16 at the fixed point. It is also shown that a previous Markovian approximation can be replaced by an exact summation to consistent order in perturbation theory.