Fig. 2
From: A numerical shallow-water model for gravity currents for a wide range of density differences
Analytical solutions of h(x,t) for the dam-break problem. Here, \(Fr = \sqrt {2}\) (Benjamin 1968) is used. a ρ c/ρ a=1.01. b ρ c/ρ a=1000. In (a), the currents at t=0.5, 1.0, 1.5, and 3.0 are in the slumping stage, and the current at t=5.0 is in the self-similar stage. The initial backward-propagating rarefaction wave arising from the rapidly removed dam travels toward the back wall x=0 (see the profile at t=0.5), reaching the wall at t=1.0. Then, a wave arises from the reflection of the rarefaction wave and travels toward the front (see the profiles at t=1.5 and 3.0). After the front is caught by this reflection wave, the current is in the self-similar stage (see the profile at t=5.0). In (b), all the currents are in the slumping stage. In this case (i.e., ρ c/ρ a=1000), the slumping stage continues until t∼226 (see Hogg 2006 for details)