Average wave overtopping discharges for complex geometry

A case study for the prediction of average wave overtopping discharges with stepped revetment in the cross-section

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Witteveen+Bos as part of construction consortium Nautilus, has prepared the design of a sea defence system in Middelkerke. The very shallow water conditions (h/Hm0 = 0.3) and complex geometry of the structure, i.e. a high berm and stepped revetment, caused uncertainties in the usage of the empirical equations from the EurOtop Manual (2018) for wave overtopping discharges. Therefore, small-scale experiments have been performed to asses the overtopping discharges.
Physical model experiments can be used to determine wave overtopping discharges, but can be expensive and time consuming. Recent studies show promising results for the use of SWASH as tool to predict overtopping discharges for relative simple geometries, but SWASH is suspected to be less accurate for more complex cross-sections. Two methods (SWASH, EurOtop) are used to predict wave overtopping discharges for the new boulevard Middelkerke and the results are compared with measurements from small-scale experiments conducted in Ghent. The goal is to asses the feasibility of using both methods for this complex structure with stepped revetment in shallow water conditions.

The EurOtop Manual (2018) uses influence factors to account for roughness at the slope and the presence of a berm in the structure. To use the empirical formula to determine the average overtopping discharge for this specific complex cross-section, a berm influence factor was added to the equation
for shallow water conditions developed by Altomare et al. (2016). Recent studies of Schoonees et al. (2021) found that the influence of the roughness of a stepped revetment on overtopping discharge mainly depends on: a characteristic step height, relative overtopping discharge and the wave period at the toe of the structure. Using their research as guideline, a roughness influence factor of γf =
0.75-0.9 was estimated for the stepped revetment for this case study. Although not validated for this specific configuration, using the (adjusted) equation resulted in very comparable average overtopping discharges compared to the physical experiments in Ghent.

In this case study, the usability of SWASH to predict the average wave overtopping discharges has been evaluated. First, the SWASH model was calibrated based on the incident wave conditions at the toe of the structure (Hm0, Tm-1,0) with the data the physical experiment in Ghent. This were the target wave conditions of this study since they are generally used to determine the average wave overtopping discharges (e.g. EurOtop) and to exclude the wave-structure interactions. Thereafter, the structure was added to the bathymetry with a smooth slope and local friction was added to represent the stepped revetment, as resolving the small steps would require an overly fine grid.

SWASH was able to reproduce the target incident wave conditions of the physical experiment very well (Hm0<3%,Tm-1,0<5%). However, when the structure was added to the bathymetry, larger differences were seen for the wave conditions at the toe compared to the physical experiment, for which no explicit explanation was found. Contrarily, Hm0 and Tm-1,0 include the full wave field, primary waves and infragravity waves, and the limited number of overtopping waves made it difficult to asses the influence of the difference in wave spectrum on the overtopping discharges.
The overtopping reduction due the added roughness/friction compared to a smooth slope reference test, was eventually related to a Nikuradse roughness height for the stepped revetment. A Nikuradse roughness of 1-1.5 times the characteristic step height of the stepped revetment, resulted in very similar average overtopping discharges compared to measurements from the Ghent experiment.

Both the EurOtop Manual (2018) and SWASH were able to reproduce the average wave overtopping discharges from physical experiments, but the results need a wide confidence band since the number of overtopping waves were limited and discharges small (q < 1 l/s/m). Each method has their own benefits and limitations. The EurOtop Manual (2018) is easy to use, but the validity of the empirical equations are uncertain for more complex geometries. SWASH gives more detailed information than only an average q: spatially and temporarily varying surface elevations and velocities along the domain, by which direct attack on the slope and structures (extreme values of u and F) can be calculated. SWASH could also be used as a ’numerical laboratory’ to further parameterize the influence of roughness on wave overtopping for a wide range of boundary conditions and structural configurations.