Energy balances for gravity currents with a jump at the interface produced by lock release

The exchanges of energy for an inviscid Boussinesq gravity current which is released from a lock and then propagates over a horizontal boundary are considered. In this study the depth ratio of ambient to lock, H, is smaller than 2; it is well known that in this case the interface of the current displays a backward-moving jump during the initial stage of the flow. The effects of this discontinuity on the energy balances are investigated using the two-layer shallow-water formulation. This is an extension of the work of Ungarish (Acta Mech 201:63–81, 2008) which covered only the smooth-interface case H ≥ 2. The energies in the current and in the ambient are calculated from numerical solutions of the initial-value shallow-water problem. Corresponding analytical energy balances are presented and the contribution of the backward-moving jump is elucidated. Both the numerical and analytical results indicate that, in general, the increase of kinetic energy of the shallow-water system cannot fully recover the decay of potential energy during the propagation. This imbalance (per unit time) of energy in the present lock-release time-dependent problem is well approximated by the rate of “dissipation” predicted by Benjamin’s (J. Fluid Mech 31:209–248, 1968) classical analysis of the steady-state current. The backward-moving jump enhances the dissipation when H is close to 1; however, for H > 1.4 (approximately) the contribution of this discontinuity to dissipation is insignificant. The general trend is that all shallow-water currents produced by lock-release are dissipative, and the dissipation increases with H. Three Froude-number formulas for the front-speed condition were considered, and it is concluded that Benjamin’s classical function is the most consistent with the motion of the jump, and produces the least dissipative currents.