Exact Solution of the Dam-Break Problem for Constrictions and Obstructions in Constant Width Rectangular Channels

Abstract

In hydraulic engineering, it is common to find geometric transitions where a channel is not prismatic. Among these geometric transitions, constrictions and obstructions are channel reaches in which a cross-section contraction is followed by an expansion. These nonprismatic reaches are significant because they induce rapid variations of the flow conditions. In the literature, the characteristics of the geometric transitions have been well studied for the case of the steady-state flow, but less attention has been dedicated to the unsteady flow conditions. The present paper focuses on the exact solution of the dam-break problem in horizontal frictionless channels where constrictions and obstructions are present. In order to find this solution, the geometric transition is assumed to be short with respect to the channel length, and a stationary solution of the shallow water equations is used to describe the flow through the nonprismatic reach. The mathematical analysis, carried out with the elementary theory of the nonlinear hyperbolic systems of partial differential equations, shows that the dam-break solution always exists and that it is unique for the given initial conditions and geometric characteristics of the problem. The one-dimensional mathematical model proves to be successful in capturing the main characteristics of the flow immediately outside the geometric transition, in comparison with a two-dimensional numerical model. The exact solution is then used to reproduce a set of experimental dam-break results, showing that the one-dimensional mathematical theory agrees with the laboratory data when the flow conditions through the constriction are smooth. The exact solutions presented here allow the construction of a class of benchmarks for the one-dimensional numerical models that simulate the flow propagation in channels with internal boundary conditions.