Turbulent entrainment in stratified flows

TLDR

When a lighter fluid is emitted under a sloping roof (or a heavier fluid from a sloping floor), it can form a thin turbulent layer whose motion is governed by entrainment of ambient fluid. The study develops a theory that entrainment is proportional to the layer’s velocity times an empirical function of the Richardson number, predicting that the layer rapidly reaches an equilibrium Richardson number and that experiments will determine this function. Two laboratory experiments—measuring the spread of a lighter surface jet and the flow of a heavier liquid down a sloping floor—were performed to determine the entrainment function E(Ri) and to estimate the ambient velocity required to reverse the layer’s motion. The experiments show that E(Ri) falls sharply with increasing Ri, becoming negligible above about 0.8, allowing flow velocities to be predicted from the supplied density difference.

Abstract

When a fluid which is lighter than its surroundings is emitted by a source under a sloping roof (or a heavier fluid from a source on a sloping floor), it may flow as a relatively thin turbulent layer. The motion of this layer is governed by the rate at which it entrains the ambient fluid. A theory is presented in which it is assumed that the entrainment is proportional to the velocity of the layer multiplied by an empirical function, E(Ri), of the overall Richardson number for the layer defined by Ri = g(ρa - ρ) h/ρaV2. This theory predicts that in most practical cases the layer will rapidly attain an equilibrium state in which Ri does not vary with distance downstream, and the gravitational force on the layer is just balanced by the drag due to entrainment together with friction on the floor or roof.Two series of laboratory experiments are described from which E(Ri) can be determined. In the first, the spread of a surface jet of fluid lighter than that over which it is flowing is measured; in the second, a study is made of the flow of a heavy liquid down the sloping floor of a channel. These experiments show that E falls off rapidly as Ri increases and is probably negligible when Ri is more than about 0·8.The theoretical and experimental results allow predictions to be made of flow velocities once the rate of supply of density difference is known. An estimate is also given of the uniform velocity which the ambient fluid must possess in order to cause the motion of the layer to be reversed.