Modelling of the upstream propagation of coastal surface waves into open channels - Application to the dynamics of undulating bores

When long waves propagate into shallow water, their dynamics changes and an undular bore may generate. This undular motion is strictly dependent on the shape of the channel in which the waves are travelling. The code developped in the CARDAMOM team at Inria makes use of a two-dimensional, depth-averaged model which accounts for non-linear and dispersive effects for waves in shallow water. First the code is used to simulate and compare the flow in a rectangular and in a trapezoidal channel. The presence of sloping banks triggers different dynamics along the cross-section of the channel and a transition of the wavelength for a certain Froude number Frt1 is observed. The simulation results are validated against experimental data. As a proof of concept, we then vary the inclination of the banks of the trapezoidal channel in order to analyse its influence on Frt1. As the numerical results show, a smaller banks slope induces longer waves and the transition takes place at higher Froude numbers. The linearised theory for long waves proposed in [Chassagne, 2017] is endorsed for gently sloping banks channels.