The solution of the Riemann problem in rectangular channels with constrictions and obstructions

Usually, the rapid geometric transitions that are of negligible length with respect to the channel are treated in one-dimensional Saint Venant models as internal boundary conditions, assuming that an instantaneous equilibrium is attained between the flow characteristics through the structure and the flow characteristics in the channel. In the present paper, a different point of view is assumed by considering rapid transients at channel constrictions and obstructions that are caused by the lack of instantaneous equilibrium between the flow conditions immediately upstream and downstream of the structure. These transients are modelled as a Riemann problem, assuming that the flow through the geometric transition is described by a stationary weak solution of the Saint Venant equations without friction. For this case, it is demonstrated that the solution of the Riemann problem exists and it is unique for a wide class of initial flow conditions, including supercritical flows. The solutions of the Riemann problem supplied by the one-dimensional mathematical model compare well with the results of a two-dimensional Shallow Water Equations numerical model when the head loss through the structure is negligible. The inspection of the exact solutions structure shows that the flow conditions immediately to the left and to the right of the geometric discontinuity may be very different from the initial conditions, and this contributes to explain the numerical issues that are reported in the literature for the rapid transients at internal boundary conditions in finite difference models. The solution of the Riemann problem has been coded, and the corresponding exact fluxes have been used as numerical fluxes in a one-dimensional Finite Volume scheme for the solution of the Shallow water Equations. The results demonstrate that spurious oscillations and instability phenomena are completely eliminated, ensuring the robustness of the approach. In the case that the energy loss is not negligible, the exact solutions capture the essential features of the two-dimensional model numerical results, ensuring that the mathematical procedure is generalizable to realistic conditions. This generalization is presented in the final part of the paper.