The Critical Richardson Number and the Ratio of the Eddy Transport Coefficients Obtained from a Turbulence Closure Model

Abstract

A turbulent closure model is analyzed under the condition that the turbulent flow is steady in its ensemble average and both the advection and diffusion terms, i.e., third moments of turbulence, are neglected in the turbulent Reynolds stress and heat flux equations. The critical flux Richardson number is defined as a limiting value beyond which physically correct solutions are no longer possible. All the turbulence moments are suppressed completely when the Richardson number exceeds the critical value. The validity of making such an assumption is tested against the numerical results which were obtained by utilizing a more complete set of equations. The critical flux Richardson numbers of 0.18σ0.27 are obtained from the different proposed empirical constants. The ratio of the eddy transport coefficient of heat to that of momentum have values of 0.5σ1.2 at the critical condition of stability. A review is made to clarity the differences between the present model and the earlier works of Ellison, Townsend, and Arya.