Two-layer shallow-water dam-break solutions for non-Boussinesq gravity currents in a wide range of fractional depth

摘要

We consider 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*. The system contains heavy and light fluids, of densities pH and pL, respectively. When the Reynolds number is large, the resulting flow is governed by the parameters R = pL/pH and H = H*/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 surface) current and the 'heavy' (bottom) current. We use a shallow-water two-layer formulations. We show that 'exact' solutions of the thickness and speed of the current and ambient can be obtained by the method of characteristics. However, this requires a careful matching with the conditions at the front (ambient) and the back (reservoir). We show that a jump, instead of a rarefaction wave, propagates into the reservoir when H < H _(crit)(R), and solutions for these jumps are presented and discussed. The theory is applied to the full-depth lock-exchange H = 1 problem, and the results are compared with previous hydraulic models. The application of the theory is also illustrated for more general cases H > 1, including comparisons with the one-layer model results of Ungarish (J. Fluid Mech., vol. 579, 2007, p. 373). Overall, the shallow-water two-layer theory yields consistent, self-contained, and physically acceptable analytical solutions for the dam-break problem over the full physical range of the R and H parameters, for both light-into-heavy and heavy-into-light gravity currents. The solution can be closed without adjustable constants or predetermined properties of the flow field. The thickness solution is formally valid until the jump, or rarefaction wave, hits the backwall; the speed of propagation prediction is valid until this reflected wave hits the nose, i.e. until the end of the slumping stage. This theory is a significant extension of the Boussinesq problem (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.