A dynamical model for turbulence. I. General formalism

Abstract

We propose new dynamical equations to describe fully developed turbulence. We begin with the Wyld equations (WE), which are exact solutions of the NSE. The WE, and their Langevin-like representation, show that nonlinearities induce a turbulent force ft(k) and a turbulent viscosity νt(k), which are given by an infinite series of Wyld diagrams. The series for νt(k) is renormalizable, and its sum can be found using RNG methods. The result, Eq. (2a), holds for stirring forces fext with an arbitrary correlation function φ and generalizes previous RNG results, which neglected ft and were limited to power law φ∼k1−2ε. To recover Kolmogorov law, these earlier RNG-based theories were forced to introduce an ad hoc stirring force with a prescribed φ∼k−3. By contrast, we show that ∼k−3 belongs to φ̃, which is the correlation function of ft, and that in the inertial range ft≫fext. The series for φ̃ cannot be summed because of a nonrenormalizable infrared divergence (IR) with an infinite number of divergent irreducible diagrams. To overcome this difficulty, we use the well-accepted notion of local energy transfer and we derive an expression for the energy flux Π(k), Eq. (2d), as well as a dynamical equation for the energy spectrum E(k), Eq. (2b). We also construct the dynamical equations for Reynolds stress spectra (solved in papers II and III). An analogous approach is developed for the temperature field. The model contains no free parameters. Some of its predictions are Kolmogorov spectrum E(k)∼k−5/3 with Ko=5/3, in agreement with recent data; temperature spectrum in the inertial-convective region Eθ∼Ba ε̄−1/3εθk−5/3, in agreement with the data; Batchelor constant Ba=σt Ko. In addition, in papers II and III we carry out extensive comparisons with the laboratory, DNS, LES data, and phenomenological models. The model can be used to construct a subgrid model for LES calculations.