A Simple Diffusion-Controled Model of Mixing Across a Stable Density Interface

Mixing across a stable density interface caused by a shear stress externally acting on a two-layer fluid initially at rest is modelled using the turbulent-diffusion concept. The influence of a (relatively weak) longitudinal pressure gradient is also considered. The central point of view developed is that the mixed layer can be only weakly stratified so that the buoyancy transport across the mixed layer, and not the processes near the interface, controls the entrainment rate. To model the turbulent transports of buoyancy and momentum, common expressions for gradient transport in turbulent Couette and channel flow are adopted. Using a similarity solution, results are given for small and large Ri, where Ri is the overall Richardson number based on the friction velocity. The entrainment rate obtained does not depend on Ri at small Ri, and is inversely proportional to Ri at large Ri. The latter result is derived without introducing the usual assumption that the increase in potential energy is proportional to the work done by the shear stress, which assumption leads to the same result. An adverse pressure gradient is found to decrease the entrainment rate. An estimate of the mean velocity of the mixed layer is given.Hydraulic EngineeringCivil Engineering and Geoscience