Gravity currents with tailwaters in Boussinesq and non-Boussinesq systems: two-layer shallow-water dam-break solutions and Navier-Stokes simulations

摘要

We consider the dam-break initial stage of propagation of a gravity current of density ρ_c released from a lock (reservoir) of height h0 in a channel of height H. The channel contains two-layer stratified fluid. One layer, called the "tailwater," is of the same density as the current and is of thickness h_T (< h0), and the other layer, called the "ambient," is of different density ρ_a. Both Boussinesq (ρ_c/ρ_a ≈ 1) and non-Boussinesq systems are investigated.By assuming a large Reynolds number, we can model the flowwith the two-layer shallow-water approximation. Due to the presence of the tailwater, the "jump conditions" at the front of the current are different from the classical Benjamin formula, and in some circumstances (clarified in the paper) the interface of the current matches smoothly with the horizontal interface of the tailwater. Using the method of characteristics, analytical solutions are derived for various combinations of the governing parameters. To corroborate the results, two-dimensional direct numerical Navier-Stokes simulations are used, and comparisons for about 80 combinations of parameters in the Boussinesq and non-Boussinesq domains are performed. The agreement of speed and height of the current is very close.We conclude that the model yields self-contained and fairly accurate analytical solutions for the dam-break problem under consideration. The results provide reliable insights into the influence of the tailwater on the propagation of the gravity current, for both heavy-into-light and light-intoheavy motions. This is a significant extension of the classical gravity-current theory from the particular h_T = 0 point to the h_T > 0 domain.