Axisymmetric gravity currents in two-layer density-stratified media

Abstract

Numerous studies have considered the flow of a rectilinear, high Reynolds number, Boussinesq gravity current through a two-layer stratified ambient, however, far less is known concerning the analogue axisymmetric problem. Whereas in both instances there is the possibility of a dynamic coupling between the gravity current front and the waves that are excited by its forward advance, axisymmetric gravity currents entail the added complexity of a radially-diverging flow. Because a steady-state formulation cannot then be developed, we instead present a one-layer shallow water model that describes the flow evolution for various initial conditions and ambient stratifications. We also report upon $>\!\!30$ full- and partial-depth lock release laboratory experiments that span a densitometric range $0 \le S < 0.8868$ where $S=(\rho _1-\rho _2)/(\rho _c-\rho _2)$ in which $\rho _c$, $\rho _1$ and $\rho _2$ denote, respectively, the densities of the gravity current and lower and upper ambient layers. Of principal interest is the initial front speed of the gravity current for which good agreement is observed between laboratory measurement and shallow water numerical simulation, despite the limiting assumptions of the latter. The horizontal distance over which the initial front speed is maintained may span several lock-lengths, however, this depends on whether or not the gravity current is substantially impacted by the interfacial wave(s). For example, when the lower ambient layer is moderate and $S$ is large, the transfer of momentum from the gravity current front to the wave may lead to a deceleration so severe that gravity current fluid is swept in the $-r$ direction. The connection between our analysis and problems of pollution dispersion is briefly outlined.

Experimental data

Tables 1 through 4 present the experimental parameters pertinent to the measured data discussed in §4. Consistent with the discussion of §2, $h_0$, $r_0$ and $r_v$ are dimensional quantities, the latter of which is determined from (2.2). The associated Reynolds number is defined by (2.1). In all cases, fluid densities were measured using an Anton Paar 4500 densitometer having an accuracy of $\pm 0.000005$ g/cm$^3$.

Table 1 Laboratory experimental data for $\Xi =1$

Table 2 Laboratory experimental data for $\varphi =0.75$ and $\Xi =2.51$

Table 3 Laboratory experimental data for $\varphi =0.5$ and $\Xi =1.46$

Table 4 Laboratory experimental data for $\Xi =1$ and $S=0$