Numerical modelling of nearshore wave transformation, breaking and overtopping of coastal protections with the enhanced Serre–Green–Naghdi equations

Admissible average overtopping discharges in given storm conditions are typically used to design coastal protections, in particular dykes or breakwaters. These discharges are usually estimated using semi-empirical formulas relying on wave conditions at the toe of the structure. These formulas, unfortunately, only work for simple configurations, invariant alongshore, and can be insufficient for complex sea states. Therefore, numerical modelling could be a more flexible alternative for estimating these discharges. This work presents the development and validation of a Boussinesq-type numerical model solving the fully-nonlinear weakly-dispersive enhanced Serre–Green–Naghdi equations for the simulation of random wave overtopping over impermeable structures in one horizontal dimension. Wave breaking is modelled with an eddy viscosity approach based on the turbulent kinetic energy, which is robust and accurate at describing energy dissipation in the surf zone. Two distinct experimental datasets, with 184 trials in total and very dissimilar wave conditions and foreshore seabed profiles, are used to validate the model regarding wave propagation, shoaling, breaking and overtopping. Both unimodal and bimodal sea states are considered. Average overtopping discharges in configurations with deep and very shallow foreshores, as well as for breaking and non-breaking waves, are well reproduced by the model. For instance, typical mean relative errors on the simulated mean overtopping rates are found to lie within $\pm 20\text{\%}$ compared with the measurements, at least for the largest discharges of the considered campaigns. The scatter of simulated discharges is somewhat higher for lower discharges, but the results remain in an acceptable range.