Gravity currents and related phenomena

Abstract

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.