Individual overtopping events are important variables when designing a coastal structure as they can deviate significantly from the mean overtopping discharge. Thus, in this study, extreme overtopping events at rubble mound structures with a smooth crest in shallow water have been studied. Both the water layer thickness (flow depth), front velocity and individual overtopping volumes are measured in a wave flume for typical coastal structures with a smooth crest in shallow water for a large range of hydraulic conditions and three different foreshore slopes. An analysis of the individual overtopping volumes shows that the largest individual overtopping volumes arise from short waves that travel on the crest of a low-frequency wave in shallow water and short waves that travel on top of the trough in deep water. Due to the temporal water level variation caused by the low-frequency waves in shallow water, there are fewer overtopping events compared to deep water conditions with the same non-dimensional overtopping discharge. However, the individual overtopping volumes of these events are larger. To quantify the extreme overtopping variables, an empirical formulation based on the relative crest height and short-wave steepness is proposed for the non-dimensional 2 % exceedance water layer thickness, front velocity and individual overtopping volume in terms of incident waves with an R² of 0.84, R² of 0.55 and R² of 0.85 respectively. A further small improvement is found when the low-frequency wave height and 2% exceedance wave height are included, but the added value of this expression does not outweigh the additional wave variables needed for the expression. A log-normal distribution with a constant shape and an expression for the scale of the distribution is proposed to describe the distribution of the individual overtopping volumes in shallow water which accurately captures the distribution (R² of 0.90). Compared to most of the current design approach which is based on a cascade of empirical formulations, this is a significant improvement. In addition, the reasonable results for a distribution with a constant shape parameter show that the shape of the distribution does not change significantly for shallow water conditions.

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Wave overtopping of coastal structures has been studied using physical model experiments with rubble mound breakwaters in shallow water. The mean overtopping discharge is determined for three different foreshore slopes and various hydrodynamic conditions. The hydrodynamic results confirm that energy is transferred to low-frequency waves in very shallow water and that the short waves are in phase with the lower-frequency waves in very shallow water. As a result, the extreme waves (e.g. 2% exceedance wave height) become relatively large in very shallow water due to the energy of the low-frequency waves affecting thereby the wave overtopping. To estimate the amount of energy at the low-frequency waves, an expression is derived which reasonably accurately predicts the low-frequency wave energy (RMSE of 0.06). Considering the non-dimensional overtopping discharge, the existing formulations for the non-dimensional mean wave overtopping discharge perform poorly to reasonably in shallow water with RMSLE ranging from 1.04 to 2.92. A parameter sensitivity study shows that the short-wave steepness, relative crest height and the low-frequency wave height are the most important parameters when predicting the mean overtopping discharge in shallow water. When including the short-wave steepness and relative crest height in an empirical formulation the RMSLE for the current dataset reduces to 0.69. A further increase in accuracy is found when the low-frequency wave height and 2% exceedance wave height are included (RMSLE 0.64).

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When constructing land reclamations, often sand is placed on top of the coarse rock of the bund surrounding the reclamation. The use of a geometrically open filter between the interface of sand and rock could be cost effective. It is expected that even a geometrically open filter with sand on top of gravel might be stable due to the arching mechanism. For such a “reversed” open filter the actual stability is unknown. Hence this study focusses on the stability of a reversed geometrically open filter under cyclic loading. This paper mainly describes the development of the test setup. First the numerical model OpenFOAM was used to extract the gradients from a representative case study. Next a test setup was developed to generate these low-magnitude loads at full-scale. Various sand-filter combinations were tested, with a range of ratios of the diameters of the gravel filter (D15F) and the sandy base layer (D85B) and sand with a unimodal distribution. They were tested for both parallel (i//) and perpendicular (iꞱ) gradients. The order of magnitude of the occurring gradients obtained with the numerical model for the case-study were a parallel gradient of i//,2% ≈ 1%, decreasing to 0 going downward, and a rather constant perpendicular gradient of iꞱ,2% = 0.2-0.3 for the lowest 4 m of the reversed granular filter. The critical perpendicular gradients were estimated at iꞱc ≈ 0.2 to 0.1 for filter ratios of D85F/D15B = 7.5 to 9.5. The critical parallel gradients were measured at i//c ≈ 2% down to 1%, but might be influenced by simultaneously occurring perpendicular gradients. Even though for the test case no stable situation could be proven with respect to the perpendicular gradient, realistic situations with stable reversed open filters seem possible.@en