Buoyancy Fluxes in a Stratified Fluid

Direct numerical simulations of the time evolution of homogeneous stably stratified shear flows have been performed for mean flow Richardson numbers in the range from 0.075 to 1.00 and for Prandtl numbers ranging from 0.1 to 2. The local or instantaneous results indicate that when the turbulent Froude number FrT = 1 the peak value of mixing efficiency is Rf ≈ 0.25, a result independent of the Prandtl number Pr and thus inconsistent with previous laboratory measurements at values of the Pr of 0.7 and 700. The results are consistent, however, with previous laboratory observations in demonstrating that the mixing efficiency Rf decreases rapidly away from this peak value at FrT = 1. Using data from both numerical simulations and laboratory experiments with both shear and grid generated turbulence, simple empirical relationships are developed to estimate Rf, and hence buoyancy flux b, for the entire range extending from active turbulence down to the limit where the effects of viscosity and/or density stratification suppress the vertical buoyancy flux.