On simple models for gravity currents from moving sources

Abstract

The propagation of the gravity current generated from a moving source of buoyancy is of interest in deep-sea mining and related technologies. The study by Ouillon et al. (J. Fluid Mech., vol. 924, 2021, A43) elucidated some salient patterns of the flow concerning a source close to the bottom on the basis of direct numerical simulation on a supercomputer. Here, we present a simple box model that provides further insights and useful analytical approximations for this gravity-current flow system. We show that this flow is very different from that produced by a moving source at the top, studied by Hogg et al. (J. Fluid Mech., vol. 539, 2005, pp. 349–385). The model confirms that the main governing parameter is the ratio $a$ of speed of source to that of buoyancy propagation. The model points out dependency also on the front-jump Froude number (which implies dependency on the height of the ambient fluid). For a sufficiently large $a >a_{crit}$ , a supercritical regime appears in which the gravity current forms a wedge behind the moving source; in the subcritical regime, the upstream propagation attains a maximum $x_m$ at time $t_m$ . The model predicts the value $a_{crit}$ , the distance and time $x_m$ and $t_m$ in the subcritical case, and the shape of the wedge in the supercritical case, without any adjustable constant. Comparisons with the numerical data show fair agreement.