Turbulence Closure, Steady State, and Collapse into Waves

Abstract

A new simple two-equation turbulence closure is constructed by hypothesizing that there is an extra energy sink in the turbulent kinetic energy (k) equation representing the transfer of energy from k to internal waves and other nonturbulent motions. This sink neither contributes to the buoyancy flux nor to dissipation, the nonturbulent mode being treated as inviscid. The extra sink is proportional to the squared ratio between the turbulent time scale τ ? k/ε, with turbulent dissipation rate ε, and the buoyancy period T = 2π/N. With a focus on high?Reynolds number, spatially homogeneous, stably stratified shear flow away from boundaries, the turbulence is described by equations for a master length scale L ? k3/2/ε and the master time scale τ. It is assumed that the onset of the collapse of turbulence into nonturbulence occurs at τ = T. The new theory is almost free of empirical parameters and compares well with laboratory and numerical experiments. Most remarkable is that the model predicts the turbulent Prandtl number, which is generally σ = σ0/[1 ? (τ/T)2], with σ0 = 1/2, and hence is not a unique function of mean flow variables. Only in structural equilibrium (τ? = 0) is the Prandtl number a unique function of the gradient Richardson number Rg: σ = σ0/(1 ? 2Rg). These forms of the Prandtl number function immediately determine the flux Richardson number Rf = Rg/σ. Steady state occurs at Rsg = 1/4 with Rf = 1/4, and within structural equilibrium the collapse of turbulence is complete at Rg = 1/2.