Exact solution of the dam-break problem for constrictions and obstructions in constant width rectangular channels

In hydraulic engineering, it is common to find geometric transitions where the channel is not prismatic. Among the other geometric transitions, the constrictions and the obstructions are channel reaches where a cross-section contraction is followed by an expansion. These non-prismatic reaches are important because they induce rapid variations of the flow conditions. In the literature, the characteristics of the geometric transitions are well studied for the case of steady state flow, while 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 non-prismatic reach. The mathematical analysis, carried out with the elementary theory of the non-linear hyperbolic systems of partial differential equations, shows that the dam-break solution always exists and it is unique for 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 of the geometric transition when compared 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 to construct a class of benchmarks for the one-dimensional numerical models that simulate the flow propagation in channels with internal boundary conditions