The exact solution to the Shallow water Equations Riemann problem at width jumps in rectangular channels

Abstract

Riemann problems at geometric discontinuities are a classic and fascinating topic of hydraulics. In the present paper, the exact solution to the Riemann problem of the one-dimensional (1-d) Shallow water Equations at monotonic width discontinuities is completely determined for any initial condition. This solution is based on the assumption that the relationship between the states immediately to the left and to the right of the discontinuity is a stationary weak solution of the 1-d variable-width Shallow water Equations. Under this hypothesis, it is demonstrated that the solution to the Riemann problem always exists, although there are cases where the solution is triple. This proves that it is possible to define width-jump interior boundary conditions of Saint Venant models that lead to well-posed problems, and that additional physical information is required to pick the relevant Riemann wave configuration among the alternatives when the solution is multiple. The analysis of an existing Finite Volume numerical scheme from the literature, based on flow variables reconstruction with preservation of specific energy and discharge, shows that the algorithm captures the solution with supercritical flow through the width discontinuity when multiple solutions are possible. This suggests that it is possible to change the algorithm accordingly to the structure of the physically relevant solution. Interestingly, the 1-d variable-width Shallow water Equations are formally identical to the 1-d Porous Shallow water Equations, implying that the exact solutions provided in the present paper are relevant for two-dimensional (2-d) Porous Shallow water numerical models aiming at urban flooding simulations.The procedure presented in this paper may be used not only to construct new challenging benchmarks for numerical schemes, but also as a guide for the construction of new algorithms, and for the interpretation of on-field and laboratory data related to transients at rapid channel width variations. The appearance of multiple solutions, which is connected to the hydraulic hysteresis phenomenon observed in the case of supercritical flows impinging a cross-section contraction, requires a criterion for the disambiguation of solutions. This will be the object of future research.