A model for fully developed turbulence

Abstract

A model for stationary, fully developed turbulence is presented. The physical model used to describe the nonlinear interactions provides an equation for the turbulent spectral energy function F(k) as a function of the time scale for the energy fed into the system, n−1s. The model makes quantitative predictions that are compared with the following available data of a different nature. (a) For turbulent convection, in the case of a constant superadiabatic gradient and for σ≪1 (σ≡Prandtl number), the convective flux is computed and compared with the result of the mixing length theory (MLT). For the case of a variable superadiabatic gradient, and for arbitrary σ, as in the case of laboratory convection, the Nusselt number N versus Rayleigh number R relation is found to be N=AσR1/3 as recently determined experimentally. The computed Aσ deviates 3% and 8% from recent laboratory data at high R for σ=6.6 and σ=2000. (b) The K–ε and Smagorinsky relations. Four alternative expressions for the turbulent (eddy) viscosity are derived (the K–ε and Smagorinsky relations being two of them) and the numerical coefficients appearing in them are computed. They compare favorably with theoretical estimates (the direct interaction approximation and the renormalization group method), laboratory data, and simulation studies. (c) The spectral function, transfer term, and dissipation term. The spectral energy function F(k), the transfer term T(k), and the dissipation term νk2F(k) are computed and compared with laboratory data on grid turbulence. (d) The skewness factor S̄3 is computed and compared with laboratory data. The turbulence model is extended to treat temperature fluctuations characterized by a spectral function G(k). The main results are (e) when both temperature and velocity fluctuations are taken into account, the rate ns(k), that in the first part was taken to be given by the linear mode analysis, can be determined self-consistently from the model itself; (f) in the inertial-convective range, the model predicts the well-known result G(k)∼k−5/3; (g) the Kolmogorov and Batchelor constants are shown to be related by Ba=σt Ko, where σt is the turbulent Prandtl number; and (h) in the inertial-conductive range the model predicts G(k)∼k−17/3 for thermally driven convection as well as for advection of a passive scalar, the difference being contained in the numerical coefficient in front. The predicted G(k) vs k compare favorably with experiments for air (σ=0.725), mercury (σ=0.018), and salt water (σ=9.2).