Equivalent riverbed slope in irregular channels

To capture different types of discontinuous solutions of the de Saint-Venant equations, the concept of one-dimensional augmented Shallow Water Equations (1D aSWE) is introduced. The aSWE has been demonstrated to be an effective tool for addressing spatial discontinuities such as bottom steps (LeFloch & Thanh, 2011), sudden width changes (Valiani & Caleffi, 2019) or, more generally, abrupt geometric variations in river cross sections of different geometry (Valiani & Caleffi, 2024). The present study focuses on a specific polyline cross-section channel, where the wetted perimeter is a 4-segment open polygonal parametrized by 2 lateral slopes and the transversal position and elevation of 3 vertices. The channel bottom is a ruled surface obtained by generating straight lines of different slopes (see Fig. 1). Such a channel is here called Uneven Origami Channel (UOC), and it is considered the prototype problem to define an equivalent longitudinal bed slope of the riverbed in irregular channels. The augmentation technique recently developed by the same authors (Valiani & Caleffi, 2024) is shown to be a suitable tool to define, solely on analytical basis, such an equivalent bed slope.