A model for the propagation of inertial gravity currents released from a two-layer stratified lock

Abstract

Consider the propagation of a gravity current (GC) released from a lock of length $x_0$ and height $h_0$ into an ambient fluid of height $H h_0$ and density $\rho _{o}$. The lock contains a layer of thickness $H_L h_0$ of density $\rho _L$ overlaid by a layer of thickness $(1-H_L)h_0$ and density $\rho _U$, where $\rho _{o} < \rho _U < \rho _L$ and $H_L \in (0, 1)$. Assume Boussinesq and large Reynolds-number flow. The internal stratification parameter is $\sigma = (\rho _L - \rho _U)/(\rho _L - \rho _{o})$, in the range $(0,1)$; the classical GC is $\sigma =0$. Such GCs were investigated experimentally (Gladstone et al., Sedimentology, vol. 51, 2004, pp. 767–789; Dai, Phys. Rev. Fluids, vol. 2, 2017, 073802; Wu & Dai, J. Hydraul. Res., 2019, pp. 1–14.); we present a new self-contained model for the prediction of the thickness $h$ and depth-averaged velocity $u$ as functions of distance $x$ and time $t$; the position and speed of the nose $x_N(t)$ and $u_N(t)$ follow. We derive a compact scaling upon which, for a given $H$ (height ratio of ambient to lock), the flows differ in only one parameter: $\varPsi = \{ [1 -\sigma (1 - H_L)]/[1 - \sigma (1 - H_L^2)] \} ^{1/2}$. The parameter $\varPsi$ equals $1$ for the classical GC and is larger in the presence of stratification; a larger $\varPsi$ means a faster and a thinner GC. The solution reveals an initial slumping phase with constant $u_N$, a self-similar phase $x_N \sim t^{2/3}$, and the transition at $x_V$ to the viscous regime. Comparisons with published experiments show good data collapse with the present scaling $\varPsi$, and fair-to-good quantitative agreement (the discrepancy and the stability conditions are discussed).