In this paper, a Reynolds-averaged Navier-Stokes (RANS) equations solver, interFoam of OpenFOAM®, is validated for wave interactions with a dike, including a promenade and vertical wall, on a shallow foreshore. Such a coastal defence system is composed of both an impermeable dike and a beach in front of it, forming the shallow foreshore depth at the dike toe. This case necessitates the simulation of several processes simultaneously: wave propagation, wave breaking over the beach slope, and wave interactions with the sea dike, consisting of wave overtopping, bore interactions on the promenade, and bore impacts on the dike-mounted vertical wall at the end of the promenade (storm wall or building). The validation is done using rare large¬scale experimental data. Model performance statistics are employed to quantify the ability of the numerical model to reproduce the experimental data. In the evaluation method, a repeated test is used to estimate the experimental uncertainty. The solver interFoam is shown to generally have a very good model performance rating.

Three open source wave models are applied in 2DV to reproduce a large-scale wave flume experiment of bichromatic wave transformations over a steep-sloped dike with a mildly-sloped and very shallow foreshore: (i) the Reynolds-averaged Navier–Stokes equations solver interFoam of OpenFOAM® (OF), (ii) the weakly compressible smoothed particle hydrodynamics model DualSPHysics (DSPH) and (iii) the non-hydrostatic nonlinear shallow water equations model SWASH. An inter-model comparison is performed to determine the (standalone) applicability of the three models for this specific case, which requires the simulation of many processes simultaneously, including wave transformations over the foreshore and wave-structure interactions with the dike, promenade and vertical wall. A qualitative comparison is done based on the time series of the measured quantities along the wave flume, and snapshots of bore interactions on the promenade and impacts on the vertical wall. In addition, model performance and pattern statistics are employed to quantify the model differences. The results show that overall, OF provides the highest model skill, but has the highest computational cost. DSPH is shown to have a reduced model performance, but still comparable to OF and for a lower computational cost. Even though SWASH is a much more simplified model than both OF and DSPH, it is shown to provide very similar results: SWASH exhibits an equal capability to estimate the maximum quasi-static horizontal impact force with the highest computational efficiency, but does have an important model performance decrease compared to OF and DSPH for the force impulse.

Wave overtopping is one of the key parameters for designing coastal structures: the crest level is usually determined using admissible overtopping discharges. Several formulae already exist for wave overtopping assessment that predict the average overtopping discharge per meter width of the coastal defence, generally for deep or intermediate water depths at the toe of the dike. However, the process of wave overtopping on sea dikes with shallow and very shallow foreshore is not yet fully understood. Gentle foreshores in combination with (very) shallow water conditions lead to heavy wave breaking and a significant change of the wave spectra from offshore to the toe of the dike. The wave steepness is assumed as one of the main criteria to identify cases of severe wave breaking on shallow and very shallow foreshores. For these conditions, Van Gent's formula, generally used for wave overtopping with shallow foreshores, has been implemented and validated against experimental data. It is the purpose of this paper to show that Van Gent's formula overestimates the average overtopping discharge for cases of very shallow foreshores. Moreover the existing formula cannot be applied to cases with an emergent toe. The present work therefore introduces a new “equivalent slope” concept to obtain an estimation of average wave overtopping discharges on sea dikes with shallow and very shallow foreshores. This study uses data from CLASH database and experimental campaigns, specifically carried out at Flanders Hydraulics Research (Belgium), in order to validate this approach. Results indicate that this concept shows better performance compared to other empirical formulae, which suggests that the influence of the very shallow foreshore on the average wave overtopping discharge should be included.