A shallow-water model for high-Reynolds-number gravity currents for a wide range of density differences and fractional depths

Abstract

We consider the propagation of a gravity current of density ρc from a lock of length x0 and height h0 into an ambient fluid of density ρa in a channel of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio. We obtain analytical solutions for the initial dam-break or slumping stage of propagation with constant speed, and derive explicitly the trends for small and large values of the governing parameters. This reveals the main features: (a) the heavy current propagates faster and its front is thinner than for the light counterpart; (b) the speed of the heavy current depends little on H*, while that of the light current increases with H*; and (c) in the shallow ambient case (H* close to 1) the light current is choked to move with the thickness of half-channel, while the heavy current typically moves with an unrestricted smaller thickness. These qualitative predictions are in accord with previous observations, and some quantitative comparisons with available experimental and numerical simulations data also show fair agreement. However, given the paucity of the available data, the main deficiency of the model is the unknown practical limit of applicability. For large time, t, a self-similar propagation with t2/3 is feasible for both the heavy and light currents, but the thickness profiles display differences.