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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The hydraulic stability of rock armour layers has been extensively discussed in the literature, with numerous formulae proposed for design purposes. However, limited attention has been given to armour stability under shallow water conditions, largely due to the scarcity of experimental data. This research aims to address this gap by providing new insights into the stability of rock armour layers with rubble mound breakwaters in shallow water. Hydraulic stability was determined for four different structure slopes and various hydrodynamic conditions, spanning from deep to extremely shallow water in presence of a 1V:30H foreshore. Newly experimental data were compared with existing stability formulae valid in shallow water, specifically those by van Gent et al. (2003, VG), Eldrup and Andersen (2019, EA), and Etemad-Shahidi et al. (2020, ES). Initially, the data were used to evaluate the accuracy of the original formulae. Following this, the formulae were recalibrated to account for the influence of shallow water, with data grouped according to water levels. Finally, modified versions of VG and ES formulae were developed to fit the experimental data, incorporating the effects of wave steepness to better capture shallow water dynamics.

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The crest level of seawalls is often based on estimates of the amount of wave overtopping. Methods to estimate the mean overtopping discharge have been provided in several guidelines. One of the important parameters affecting wave overtopping is the wind. However, the effects of wind have not been accounted for in detail in present design guidelines although some guidance for coastal structures with crest elements is provided in literature. For onshore wind the expected wave overtopping discharge at coastal structures with a crest element can be up to a factor 5 larger than for situations without wind. In the present study the maximum influence of wind on wave overtopping at impermeable seawalls with crest elements has been studied based on physical model tests. The result of the study is a guideline to estimate the maximum influence of wind on wave overtopping at seawalls with crest elements.

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Rising sea levels caused by climate change are increasing the risk of overtopping on coastal structures. Moreover, there is a growing societal concern about the visual impact of these structures, which leads to the lowering of their crest freeboards. In previous studies, safety during overtopping events was assessed considering the overtopping layer thickness (h_c), the overtopping flow velocity (u_c) and the individual wave overtopping volume (V). Existing models in the literature to estimate h_c, u_c and V on mound breakwater crests are mainly deterministic, involve a chain of successive estimations leading to accumulated errors and/or do not account for the dependencies between h_c, u_c and V. This study proposes a model to describe the joint probability distribution of h_c, u_c and V based on bivariate copulas. Experimental data from small-scale 2D physical tests conducted on mound breakwaters with three armor layers (single-layer Cubipod®, and double-layer cubes and rocks) in depth-limited breaking wave conditions on two mild bottom slopes and dimensionless crest freeboards between 0.33 and 3.20 is used. Lognormal distribution functions are proposed for each variable and a multivariate dependence model is developed through a one-tree vine-copula. The parameters of this model are quantified directly using wave characteristics and the structure geometry minimizing the accumulated errors in the final predictions. The application of the model is illustrated by computing the probability of not fulfilling at least a tolerability limit for one of the studied variables (OR probability). The OR probability is computed both considering the dependence and assuming independence between the variables and a significant difference is obtained. It is concluded that by accounting for the multivariate dependence between the variables, it is possible to reduce the crest freeboard and, thus, achieve a more economic design within the required safety level.

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Initial damage, caused by previous wave loading or other events, might affect the hydraulic stability of pattern-placed revetments. Three common types of damage are considered in this study. The effect of this assumed initial damage on the hydraulic stability and failure probability of revetments is quantified using a FEM model. This model is developed using data from large-scale flume and field experiments. Using results from the FEM model, surrogate models are created to predict the effect of each type of initial damage on the hydraulic stability and failure probability. Through the use of these surrogate models, it is demonstrated that S-shaped deformation caused by filter migration around the wave impact zone has the largest effect on the hydraulic stability decreasing up to 30%, and failure probability per year increasing up to 10,000 times. When the granular filling between the joints of the columns is washed-out, the stability decreases up to 29% and the failure probability increases up to 700 times. A missing column has a limited effect on the hydraulic stability and failure probability when there is no other (structural) damage. However, if it originates from underlying damage, it might be an initial sign of total failure of the revetment. This study demonstrates the effectiveness of finite element modeling for studying (damaged) revetments, which can be used to complement flume experiments. The results can be used to prioritize maintenance efforts in risk-based maintenance of pattern-placed revetments.

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Physical model tests have been performed to study static stability of rock-armoured mild slopes. Current stability design formulae for steeper rock-armoured slopes focus on plunging and surging waves. Slopes of 1:6 and milder usually have more spilling breakers which decreases the load. Also, on mild slopes displaced rocks more often remain present in the wave attack zone, which increases the strength. These aspects lead to an overdesigned structure when existing formulae for steep rock-armoured slopes are used. The present wave flume tests were used to understand the processes and develop a design formula for rock-armoured mild slopes with an impermeable core. These tests were performed for statically stable rock-armoured slopes of 1:6 to 1:10. The tests confirmed that not all existing damage parameters are able to accurately describe the static stability on milder slopes. For mild slopes it is more accurate to describe the damage based on the eroded depth rather than on the eroded area or number of moved stones. In this study, a design formula and guidelines are provided for practicing engineers that design or evaluate the stability of mild rock-armoured slopes.

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During storms, ensuring the protection of people, vehicles and infrastructure on the crest of coastal structures from wave overtopping hazards is crucial. The thickness of the wave overtopping layer is a key variable used for assessing safety and maintaining a secure design. Traditionally, this parameter is associated with the height difference between the fictitious wave run-up level exceeded by 2% of waves and the crest freeboard of coastal structures. This study aims to investigate the wave overtopping layer thickness on the crest of rubble mound seawalls. To achieve this, a series of 125 small-scale 2D physical model tests were conducted on a two-layer rubble mound seawall with an impermeable core and slopes of 1:1.5 and 1:2. The obtained results indicated that the existing empirical formulas, originally developed for dikes, underestimate the overtopping layer thickness on the studied seawall. Therefore, modifications were made to the formulas found in the literature specifically tailored for rubble mound seawalls. The newly proposed formulas for estimating overtopping layer thickness at both the seaward edge and the middle of the crest showed improvements compared to the existing formulas.

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Single layer randomly placed armour units are used in many rubble mound breakwaters around the world. For these armour layers first extraction of units starts at high loads and can then progress quickly. Before the first extraction of a unit, typically no quantitative description of damage can be given. But additional to extraction, breakage of armour units due to rocking could be a major damage mechanism. This paper treat novel embedded Rocking Sensors. The technique is used to obtain the first measurements of rocking-impact velocities of single-layer units. They are also the first tests where the instrumented units can naturally move with the compacting layer during storm build-up. Physical model tests were performed on an armour layer with XBloc units. With 8 to 10 instrumented units per test run, in total 640 single measurements of the rocking motion of a unit during a 1200 wave test run were obtained, for three water levels and five wave heights. From the Rocking Sensors the number of impacts and rotational impact velocities were obtained. From an image analysis the along-slope settlement of the units during the tests was quantified. The rotational motion expressed by was found to be most convenient to express the motion. It can be seen that the units in the armour layer rock much more often than visually observed. Settlement seems to be a continuous progress, with most units rocking intermittently. Highest impact velocities are seen to occur around the water line, and in the uprush phase of the waves. A maximum impact velocity for all tests of 0.34 m/s (model scale) was measured. A preliminary design expression for rocking impact velocities of single layer units (Xblocs) is given. The paper shows that novel measurement techniques like the Rocking Sensors and vision techniques can and should be used to quantify damage mechanisms to rubble mound single-layer armour, additional to counting the extracted number of intact units.@en

Randomly placed breakwater armour units under wave loading can sometimes start rocking, which can lead to breakage of armour units. This failure mechanism can especially become important for single layer randomly placed armour units for which full displacement of units will only happen at higher stability numbers compared to older types of units, and where unit breakage can more easily lead to progressive damage to the armour layer. However, unlike older types of units, hardly any quantitative information is available on the impact velocities, and the number of impacts is mostly assessed using somewhat subjective visual observations. In design the observed number of rocking units is limited to the amount of visually observed rocking units. Hence a good quantification of impact velocities could lead to a more optimal design. This paper describes the further development of embedded rocking sensors to measure the motions of individual smart armour units. Multiple smart rocking sensors have been applied in a physical model of a breakwater and measurements were collected to determine the number of impacts and impact velocity of the armour units. The results have been compared to visual observations and the first results will be presented. It is concluded that the new technique can be used to obtain much more information on rocking, including impact velocities, and that more rocking occurs than is observed visually.@en

Slope stability formulae for rubble mound structures are usually developed for head-on conditions. Often, the effects of oblique waves are neglected, mainly because it is assumed that for oblique wave attack, the reduction in damage compared to perpendicular wave attack is insignificant. When the incident waves are oblique, the required armour size can be reduced compared to the perpendicular wave attack case. Therefore, it is important to consider the wave obliquity influence on slope stability formulae as a reduction factor. One of the most recent formulae for estimating the stability of rock-armoured slopes, referred to as Etemad-Shahidi et al. (2020), was proposed for perpendicular wave attack. The aim of this study is to develop a suitable wave obliquity reduction factor for the above-mentioned stability formula. To achieve this, first, laboratory experiment datasets from existing reliable studies were selected and analysed. Then, previously suggested reduction factors were evaluated and a suitable reduction factor for the mentioned stability formula were suggested. The suggested reduction factor includes the effect of wave obliquity and directional spreading explicitly. It is shown that the stability prediction is improved by using the wave obliquity reduction factor.

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Numerical modelling of wave interaction with rock-armoured rubble mound breakwaters has been performed to study wave overtopping. The influences of the slope angle, a berm in the seaward slope, a protruding crest wall, a recurved parapet, and the wave steepness have been studied using a validated CFD model (OpenFOAM). The numerical modelling confirms trends that have been observed in physical model tests while the validity of earlier developed guidelines has been examined outside the ranges of the physical model tests on which the guidelines are based. The numerical model results confirm that wave overtopping at rubble mound breakwaters depends on the wave steepness, that the influence of a berm is affected by the wave steepness, and that an earlier developed influence factor to account for the effects of a protruding crest wall can be applied to even larger crest walls than the tested crest walls on which the guidelines are based. The results indicate that the influence of the applied core material of the berm on the discharges is very limited. The numerical model also indicates that applying a recurved parapet on a crest wall of a rubble mound breakwater only has an effect for very small overtopping discharges. The numerical model results show that wave overtopping at rubble mound breakwaters strongly depends on the slope angle. Since this effect is so large that it cannot be neglected, while present guidelines for non-breaking waves do not include the effect of the slope angle, modified guidelines have been proposed. The observed effects of the slope on wave overtopping discharges at rubble mound structures still need to be verified based on physical model tests.

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Conventional rubble mound structures such as breakwaters, seawalls, and revetments are the most common type of coastal structures around the world used to protect harbour basins and embankments from wave action. To have a safe and economic design, two aspects need to be considered. The first one is the structural stability where the required armor size (weight) must be determined. The second aspect is the safety, where the crest freeboard of the structure is usually determined based on the allowable mean wave overtopping rate. Several semi-empirical formulas have been developed for these purposes. These formulas, which have evolved over time, are generally semi-empirical and based on the small-scale laboratory experiments where both incident wave characteristics and the structure configuration are considered. This paper aims to provide a comprehensive overview of the performance of existing formulas developed for the assessing the stability and mean overtopping rate of conventional rubble mound structures, while also introducing the recent ones. The Rock Manual formulas for the slope stability and EurOtop formula for estimating the mean overtopping rate will be discussed, and their performances will be compared with those of more recent and comprehensive ones using both lab and field data. It will be shown that the recent formulas that utilize the spectral energy mean period for stability analysis and run-up for the mean overtopping rate are more robust and physically sound. Finally, design formulas and uncertainty estimates are presented, along with guidance for practitioners.

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Wave transmission at low-crested coastal structures has been studied, based on physical model tests with trapezoidal impermeable, permeable and perforated structures. The differences between wave transmission at impermeable and permeable structures are relatively limited. For a perforated hollow structure with an impermeable vertical screen in the middle, the wave transmission is significantly less than for perforated structures without an impermeable vertical screen; the blocking of the orbital motion by the screen significantly reduces wave transmission. The effectiveness of an impermeable vertical screen to block the orbital motion and consequently reduce wave transmission assists designers of artificial reefs to design structures that reduce wave transmission. Empirical expressions based on a hyperbolic tangent function have been derived to describe the test results. For permeable structures also available data for emerged structures has been used in the analysis, and the newly introduced expression appears to be accurate for both submerged and emerged permeable structures.

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The mean overtopping discharge and the overtopping flow parameters related to individual overtopping events are often used to characterize the wave overtopping processes at dikes. Roughness, berms and oblique waves have significant effects on the wave overtopping processes at dikes while these effects are still not fully understood. A 2DV OpenFOAM® model is validated using experimental data for predicting flow velocity and layer thickness at the waterside edge of the crest. The validated numerical model is then applied to investigate the effects of roughness and a berm on the flow velocity and layer thickness. The roughness is modelled by creating protrusions along the waterside slope. Numerical model outcomes indicate that existing empirical formulas underestimate wave overtopping quantities. Introducing a roughness factor to existing empirical formulas leads to better estimates of the flow characteristics. We found that the flow characteristics are more sensitive to the variation of the berm width than to the berm level. Model results demonstrate that existing formulas for predicting the flow characteristics, as derived based on smooth straight slopes, also work well for slopes with a berm. Rayleigh and Weibull distribution functions are derived to estimate the flow velocity and layer thickness with exceedance probabilities below 10%. In order to take oblique waves into account, the 2D numerical model is extended into the 3D model domain. This 3D OpenFOAM® model is validated using measured mean overtopping discharges. The influence of oblique waves on the mean overtopping discharge in combination with a berm is analysed. The numerical model computations confirm that the reductive influence factor of oblique waves is dependant on the berm width.

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Estimation of the mean overtopping discharge is a major task in the design and assessment of the crest level of rubble mound structures such as breakwaters and seawalls. The tolerable mean overtopping rates are given based on the associated risk and wave characteristics. Several empirical formulas have been developed for the prediction of mean overtopping discharge at coastal structures. These formulas can be applied to a wide variety of coastal structures, but have limited accuracy and/or do not reflect the physics of the phenomena. The main aim of this study is to overcome these issues for rubble mound structures by considering the physics of the process in the formula development. To achieve this, first, the references used in the extended CLASH database (also called the EurOtop-2018 database), were scrutinized, the reported wave characteristics were corrected (if required) and the rubble mound structure subset was extended using a recent study. Then noting that overtopping occurs when the wave runup exceeds the freeboard, the difference between the maximum wave runup and crest freeboard was considered as the governing parameter in the mean overtopping discharge formula. In the developed formula, a semi-empirical relationship between the mean overtopping and wave runup has been established. The performances of the developed formulas and existing ones were evaluated both qualitatively and quantitively. Accuracy metrics such as RMSE and BIAS indicated the superiority of the developed simple formula. Finally, a design formula to consider uncertainty and some guidelines are provided for practitioners.

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The performance of high-density cubes in the armor layer of a breakwater was investigated through an experimental study. For this purpose, two different cross-sections, one with a conventional cross-section and one with a berm, were modelled in a wave flume. A total of ten tests were performed for both cross-sections. Different concrete densities of cubes and different placement methods were applied. The dimensions of the cubes with two different densities were the same for all tests to avoid potential scale effects caused by the dimensions of HD cubes. The experimental study showed that HD cubes were more stable than ND cubes; the characteristic wave height for HD cubes was 1.5 times higher than that of the ND cubes. The use of HD cubes may provide economic efficiency and enables a significant reduction of the concrete volume, transportation of material, and reduction of required construction area for manufacturing, etc. HD cubes in a single layer turned out to be very stable. However, only using HD cubes in a part of the armor layer or only used on the second (upper) layer caused a decrease in the stability of the armor layer, especially in comparison to the armor layers that fully consist of HD cubes. The performance of HD cubes was also evaluated by comparing the experimental results with the Van der Meer (1988) formula and the suitability of the stability number for HD cubes was discussed.

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For a safe design of a rubble mound seawall, overtopping characteristics such as the mean overtopping discharge (q) and the maximum individual overtopping volume (V_max) should be limited. Unlike q, the estimation of V_max is more complex and requires a wave-by-wave analysis of overtopping as well as a statistical analysis. The present study contributes to the knowledge of the distribution of individual overtopping volumes and the estimation of V_max at rubble mound seawalls. A total of 135, small-scale 2D physical model tests were conducted across a practical range of crest freeboards and considered the slopes of 1:1.5 and 1:2. The well-known 2-parameter Weibull and Exponential distributions were first fitted to the experimental data to estimate the V_max. Different approaches to sample the observed distribution of wave-by-wave overtopping volumes were evaluated including a threshold method using the top 10%, 30%, and 50% of individual overtopping volumes, and a method that applies a greater weighting to the larger events. For both Weibull and Exponential distributions, the weighted method was found to be the best one providing a 23% and 17% decrease in scatter index (SI) values compared to the best of existing methods. To facilitate the estimation of V_max for design purposes, a simple empirical formula was developed as a function of the dimensionless mean overtopping discharge (q*) and the number of overtopping waves (N_ow). This formula with SI = 37% outperformed the distribution-based methods as well as the best of existing formulae for V_max. In the case of the normalised bias (NBIAS), the distribution methods underestimated V_max by −21% (Weibull) and −31% (Exponential) whereas the new formula yielded NBIAS = −6%.

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Seawalls play a significant role in protecting coastal areas against wave attack and flooding. The accurate estimation of wave overtopping at seawalls is therefore crucial to adequately protect people and infrastructure in these regions. In this study, the mean wave overtopping rate at rubble mound seawalls was investigated through 140 small-scale physical model tests which adds to the limited existing data for this structure type in the extended CLASH database called EurOtop (2018). The combined dataset is used to evaluate the prediction skill of existing empirical formulae and to identify their limitations. The role of wave steepness on the mean overtopping rate is closely examined as it has not yet been considered properly in the EurOtop (2018) formulation. A new formula was derived using dimensional analysis and physical justifications of the overtopping phenomenon. The formula was found to provide a 40% decrease in RMSE in comparison to that of the EurOtop (2018). In addition, the new formula yields a BIAS ≈0, a significant improvement compared to −0.38 (non-dimensional discharge) of the EurOtop (2018) formula. The proposed formula has a simple form where non-dimensional overtopping discharge depends only on the relative crest freeboard and wave steepness, which were found to be the most important variables based on a sensitivity analysis.

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Physical model tests have been performed to study wave overtopping at rubble mound breakwaters, including breakwaters with a crest wall, breakwaters with a berm, and breakwaters with a crest wall and a berm. For rubble mound structures with a protruding crest wall or with a stable berm, limited information is available in literature even though protruding crest walls and berms clearly affect wave overtopping discharges. Adding a crest wall to an existing structure, increasing the height of a crest wall, adding a berm, or increasing the width or height of a berm, can be effective measures to account for effects of sea level rise if the sea level rise appears to be more severe than the amount of sea level rise for which the structure was designed for. The present wave flume tests were used to develop guidelines for rubble mound breakwaters, including breakwaters with a crest wall or with a berm. The relative height of the protruding part of a crest wall dominates the effect of a crest wall. The berm width, berm level and wave steepness all affect the influence of a berm on the wave overtopping discharge. Moreover, it was confirmed that the wave steepness also affects wave overtopping discharges for rubble mound breakwaters without a berm or without a crest wall. The developed set of expressions for rubble mound structures has also been validated based on existing data for oblique wave attack on rubble mound breakwaters with a crest wall.

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Coastal structures are often designed to a maximum allowable wave overtopping discharge, hence accurate prediction of the amount of wave overtopping is an important issue. Both empirical formulae and neural networks are among the commonly used prediction tools. In this work, a new model for the prediction of mean wave overtopping discharge is presented using the innovative machine learning technique XGBoost. The selection of features to train the model on is carefully substantiated, including the redefinition of existing features to obtain a better model performance. Confidence intervals are derived by tuning hyperparameters and applying bootstrap resampling. The quality of the model is tested against four new physical model data sets, and a thorough quantitative comparison with existing machine learning methods and empirical overtopping formulae is presented. The XGBoost model generally outperforms other methods for the test data sets with normally incident waves. All data-driven methods show less accuracy on oblique wave data, presumably because these conditions are underrepresented in the training data. The performance of the XGBoost model is significantly improved by adding a randomly selected part of the new oblique wave cases to the training data. In the end, this new model is shown to reduce errors on all data used in this work with a factor of up to 5 compared to existing overtopping prediction methods.

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