The Numerical Simulation of Steady and Unsteady Flows in Channel Networks Using an Extended Riemann Problem at the Junctions

In this work, an analysis of the numerical simulations of the channel flows in symmetrical and asymmetrical networks using the one-dimensional shallow water equations (SWE) was performed. Inner boundary conditions must be imposed at the junction nodes, and their numerical treatment is crucial to develop reliable mathematical models. Classical methods to provide the inner boundary conditions that are extensively based on the energy and the momentum balances at the junction nodes. However, such methods suffer their empirical formula based on experimental coefficients, which limit their applicability in practical applications (river flow, dam flow). As an alternative to such methods, we propose to solve an Extended Riemann Problem (ERP) at the junction nodes, consistently with the physical and mathematical properties of the SWE. No empirical coefficients are involved, and this approach can be easily used in practice. Furthermore, the ERP approach is supported by theoretical evidences (i.e., existence and uniqueness theorems), that ensure the consistency of the numerical scheme. This theoretical analysis can be found in literature for a star network of three identical channels, and it is extended to general network configurations by its authors. Considering subcritical conditions, different flow simulations are performed in symmetrical and asymmetrical confluences for both steady and unsteady flows. The numerical solutions are compared to experimental data from literature and analytical solutions obtained by the authors. The overall results show the suitability of the ERP method to successfully provide the inner boundary conditions for wide range of applications.