A 1D numerical model for the simulation of unsteady and highly erosive flows in rivers. (15th March 2019) - Martínez-Aranda, S.; Murillo, J.; García-Navarro, P.
Highlights: Coupled Roe's scheme for SW-Exner equations in cross-section averaged erosive flows. Explicit expressions for eigenvalues, eigenvectors, wave and source strengths. A new parameter cb related to the bed change appears in the coupled Jacobian matrix. cb depends on the erosion-deposition mechanism selected to update the cross-section. Analytical expression for the equilibrium bed slope in variable width rectangular channels. Abstract: This work is focused on a numerical finite volume scheme for the coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expressions modeling the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled equations can be rewritten as a non-conservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. Moreover, the wave celerities for the coupled morpho-hydrodyamical system depend on the erosion-deposition mechanism selected to update the channel cross-section profile. This influence is incorporated into the system solution by means of a new parameter related to the channel bottom variation celerity. Special interest is put to show that, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge in presence of arbitrary shape channels. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver, first order accurate in time and space, dealing with solid transport flux variations caused by the channel geometry changes. Channel cross-section variations lead to the appearance of a new solid flux source term which should be discretized properly. The stability region is controlled by wave celerities together with a proper reconstruction of the approximate local Riemann problem solution, enforcing positive values for the intermediate states of the conserved variables. Comparison of the numerical results for several analytical and experimental cases demonstrates the effectiveness, exact well-balancedness and accuracy of the scheme.