Dam break in power-law cross-section channels with different upstream/downstream widths

The dam break problem is studied in a channel characterised by a power-law cross-section, with different widths upstream and downstream of the dam. This research is the extension of a previous study by Valiani and Caleffi (2019), designed for rectangular cross-sections, to different and more complex geometries. A 1D a-SWE (one dimensional augmented Shallow Water Equation) system is developed, which has been shown to capture the full range of possible solutions. To address abrupt changes in the section, the classic SWE system is augmented with a third equation consisting of the time-invariance of the width scale. The idea of the augmented system was introduced by LeFloch and Thanh (2011) for the unit-width SWE, using the bed elevation as an additional variable, and by Valiani and Caleffi (2019) for rectangular narrowing/widening cross-sections, using the channel width as an additional variable. The numerical model is a Finite Volume Method, second-order accurate in space and time, using a path conservative scheme to evaluate the numerical flux at the cell interfaces. A nonlinear path is adopted, which is shown to be optimal for capturing both the sharp contact wave at the dam and the moving shock(s) downstream of the dam. The comparison of the numerical results with the analytical results for different solution patterns of the solution supports the confidence in the reliability of the model.