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 ͑WE͒, which are exact solutions of the NSE. The WE, and their Langevin-like representation, show that nonlinearities induce a turbulent force f t ͑k͒ and a turbulent viscosity t (k), which are given by an infinite series of Wyld diagrams. The series for t (k) is renormalizable, and its sum can be found using RNG methods. The result, Eq. ͑2a͒, holds for stirring forces f ext with an arbitrary correlation function and generalizes previous RNG results, which neglected f t and were limited to power law ϳk 1Ϫ2⑀ . To recover Kolmogorov law, these earlier RNG-based theories were forced to introduce an ad hoc stirring force with a prescribed ϳk Ϫ3 . By contrast, we show that ϳk Ϫ3 belongs to , which is the correlation function of f t , and that in the inertial range f t ӷ f ext . The series for cannot be summed because of a nonrenormalizable infrared divergence ͑IR͒ 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 ⌸(k), Eq. ͑2d͒, as well as a dynamical equation for the energy spectrum E(k), Eq. ͑2b͒. We also construct the dynamical equations for Reynolds stress spectra ͑solved in papers II and III͒. An analogous approach is developed for the temperature field. The model contains no free parameters. Some of its predictions are Kolmogorov spectrum E(k)ϳk Ϫ5/3 with Koϭ 5 3 , in agreement with recent data; temperature spectrum in the inertial-convective region E ϳBa ⑀ Ϫ1/3 ⑀ k Ϫ5/3 , in agreement with the data; Batchelor constant Baϭ t 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.

Turbulence in Stars. III. Unified Treatment of Diffusion, Convection, Semiconvection, Salt Fingers, and Differential Rotation

“…The Ðnal result does justify empirical models of the type shown in equation (128), with the advantage that the parameters a and b are no longer free, and in most cases they are not even constant, since they depend on the turbulent Prandtl number which itself is a function of Pe p t , (Canuto & Dubovikov 1996) :…”

“…The construction of the function has improved considerably through the use of the l t (k) RNG-based techniques. As discussed elsewhere in detail (Canuto & Dubovikov 1996 and references therein), the RNG-based model yields the following expressions for the Ðve of equation (126) :…”

Small scale response and modeling of periodically forced turbulence

“…(28) and (22). Whereas it is certainly expected that a two-equation model should be adequate in the static limit, it may be surprising that the frozen turbulence limit fork is also predicted correctly, particularly in view of the suggestion 5 that in oscillating channel flow, predicting the frozen turbulence limit requires rapid distortion theory.…”

An analytical theory of the buoyancy–Kolmogorov subrange transition in turbulent flows with stable stratification

“…The random force f β (k) is generated by nonlinear interactions as elaborated in the earlier studies [40,[44][45][46][47]. Numerous attempts to derive this force from first principles have been unsuccessful so far [40,48], and so its form needs to be postulated. Physically, f β (k) accounts for stirring of a mode k by all other modes; it is solenoidal, zero-mean, white noise in time, isotropic and homogeneous in time and space.…”