Dam-break reflection

Abstract

Summary The unsteady reflection of dam-break flow along a horizontal channel by a remote barrier is modelled using the nonlinear shallow water equations. The interaction generates an upstream moving bore that connects the collapsing reservoir of fluid to a rapidly deepening fluid layer adjacent to the barrier. These motions are modified when the fluid is released into a channel containing a pre-wetted layer, because the oncoming flow is itself headed by a bore that alters the initial reflection. Solutions for these flows are calculated using quasi-analytical techniques that utilise the method of characteristics and the hodograph transformation of the governing equations, and the results are validated by comparison with direct numerical integration of the shallow water equations. The analytical solutions enable the precise identification of dynamical features in the flow, including the onset and development of discontinuous solutions that are manifest as bores, as well as their long term behaviour, the rate at which energy is dissipated, and for flows generated from the release of a finite reservoir, the maximum depth of the fluid layer at the barrier.

A simple model for the reflection by a vertical barrier of a dambreak flow over a dry or pre-wetted bottom

“…the jump, of this stagnant fluid must move with a significant speed to accommodate the increasing volume of the arrested fluid. (2) The solutions reported by HS strongly support this expectation. For the classical dry-bottom case, figure 5( b ) of HS displays a very small in the domain between the jump and the wall.…”

“…(2) The solutions reported by HS strongly support this expectation. For the classical dry-bottom case, figure 5( b ) of HS displays a very small in the domain between the jump and the wall. For the tailwaters case, HS report a ‘first stage’ with in the relevant domain (as discussed later in § 3).…”

“…See figure 1. We assume that , the influence of the sidewalls is negligible, and the backwall of the reservoir is far away from the dam (consistent with the major analysis of Hogg & Skevington 2021) and hence the only obstruction and reflection is from the wall at . We emphasize the choice of the horizontal scale : half the distance between the dam and the obstacle (wall).…”

“…For the classical dry-bottom case, figure 5( b ) of HS displays a very small in the domain between the jump and the wall. For the tailwaters case, HS report a ‘first stage’ with in the relevant domain (as discussed later in § 3). (3) This stagnant-downstream assumption in the analysis of reflected bores from the wall of a reservoir in two-layer gravity currents has been corroborated by experiments and simulation (see Baker, Ungarish & Flynn 2020; Ungarish 2020); since there are physical similarities between the systems, we expect the same good performances in the present problem.…”

“…The mathematical advantage of this set-up is that the governing shallow-water equations are amenable to solution by the reliable method of characteristics. Such theoretical investigations were carried out by Greenspan & Young (1978) and recently revisited and extended by Hogg & Skevington (2021), referred below as HS. However, in presence of reflected jumps, the solution by characteristics is an applied-mathematical challenge.…”