The dam-break of non-Boussinesq gravity currents of various fractional depth: two-layer shallow-water results

The dam-break initial stage of propagation of a gravity current released from a lock of length $x_0$ and height $h_0$ into an ambient fluid in a channel of height $H^\ast$ is considered. The system contains heavy and light fluids, of density $\rho_H$ and $\rho_L$, respectively. When the Reynolds number is large, the resulting flow is governed by the parameters $R= \rho_L/\rho_H$ and $H = H^\ast/h_0$. We focus attention on non-Boussinesq effects, when the parameter $R$ is not close to $1$; in this case significant differences appear between the ``light'' (top) current and the ``heavy'' (bottom) current. Using a shallow-water two-layer formulations, we obtain ``exact'' analytical solutions for the thickness and speed of the current and ambient by the method of characteristics. We shown that a jump (instead of a rarefaction wave) propagates into the reservoir when $H < H_{crit}(R)$, and that propagation with critical speed occurs for some combinations of $H,R$. The theory is applied to the full-depth lock exchange $H=1$ problem, and also to more general cases $H >1$. Comparisons to previously published results are discussed. This is a significant extension of the Boussinesq problem (which is recovered by the present solution for $R = 1$), which elucidates the non- Boussinesq effects during the first stage of propagation of lock-released gravity currents.