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.

Laboratory experiments of wave propagation over rigid and flexible vegetation fields, with the same configurations, were conducted to understand the effect of vegetation flexibility on the drag coefficient (C_D). The direct method and the least squares method (LSM), based on force and flow measurements, are applied to calculate the C_D in the experimental conditions. The formulations of both methods are extended to estimate the C_D for flexible vegetation cases. A video analysis was performed to account for the swaying motion. Typically, wave dissipation is lower for flexible than for rigid vegetation of the same configuration, under the same flow condition. Therefore, a proportional effect in the corresponding C_D results, obtained from common C_D calibration to wave dissipation without considering vegetation motion, is usually observed. However, the present results show that although the wave dissipation was 34% lower for flexible relative to rigid vegetation, the respective C_D values were close. C_D estimations considering vegetation motion and inertia suggest that C_D of flexible vegetation was up to 13% higher relative to rigid vegetation. Accounting for inertia reduced the C_D for rigid vegetation up to 7%, while raised the C_D for flexible vegetation up to 13%.

Understanding key flooding processes such as wave overtopping and overflow (i.e., water flows over a structure when the crest level of the structure is lower than the water level in front) is crucial for coastal management and coastal safety assessment. In port and harbour environments, waves are not only perpendicular to the coastal structure but also very oblique, with wavefronts almost perpendicular to the main infrastructures in the harbour docks. Propagation and wave–structure interaction of such perpendicular and (very) oblique waves need to be appropriately modelled to estimate wave overtopping properly. Overflow can also be critical for estimating flooding behind any coastal defence. In this study, such oblique and parallel waves (i.e., main wave direction is parallel to the structures) are modelled in a non-hydrostatic wave model and validated with physical model tests in the literature. On top, overflow is also modelled and validated using an existing empirical formula. The model gives convincing behaviours on the wave overtopping and overflow.

Vegetation meadows in coastal waters are a key constituent of a future green defense package due to the ecosystem services they provide and the potential to attenuate wave energy. To numerically describe the vegetation dynamics under wave action, this paper presents a novel application of a numerical coupling for solving fluid–elastic structure interactions (FSI) problems involving ultra-thin elements in a 3-D environment. The extended two-way coupling employed in this work combines the mesh-free Smoothed Particle Hydrodynamics (SPH) method in the DualSPHysics code to solve the fluid flow, and the Finite Element Analysis (FEA) structural solver in Project Chrono to solve the structural dynamics. To represent the vegetation, a flexible structure based on the Euler–Bernoulli beam model is used. The beam element is embedded into the SPH domain using an envelope subdomain that is discretized using dummy boundary particles. As such, this dummy envelope serves as a decoupling interface for the geometrical properties of the structure, allowing for ultra-thin structures smaller than the initial inter-particle distance (dp). The numerical approach is validated against an experimental setup including a flexible blade swaying under the action of an oscillatory flow. The results demonstrate that the numerical model is able to resolve the wave–vegetation interaction problem. Furthermore, additional insights into the blade dynamics reveal that the swaying velocity increases linearly along the length, with the upper part swaying at a speed comparable to the fluid velocity while the stem remains relatively stationary. Additionally, the findings indicate that rigid vegetation experiences higher forces per unit length, and in systems with substantial swaying motion, energy dissipation predominantly occurs around the lower base of the vegetation.

This work presents the results of an investigation on how wave overtopping at a near-vertical seawall at the back of a sandy foreshore is influenced by sequences of erosive storms. The experiments were carried out in the Large Wave Flume (GWK) at Leibniz University, Hannover (Germany). The tested layout consisted of a near-vertical 10/1 seawall and a sandy foreshore with an initial 1/15 slope. Three sequences of idealised erosive storms were simulated. Within each storm both the incident wave conditions and still water level were varied in time to represent high and low tide conditions. Each sequence started from a 1/15 configuration and the beach was not restored in between storms. The measurements included waves, beach profile, wave overtopping volumes. The profile of the beach was measured after each sea state tested. Wave overtopping at each stage of the tested storms was significantly influenced by bed changes. This was linked to the measured evolution of the beach. Measurements showed that a barred profile developed quickly at the start of each sequence, and scour developed at the toe of the structure during high water level conditions, while accretion or partial backfilling developed during low water level conditions. Due to these processes, the position of a sea state in the tested sequence is shown to be an important factor in determining the wave overtopping volume. Remarkably, when a weaker idealised storm followed a more energetic one, nearly the same level of overtopping was recorded. This is explained by the foreshore erosion, leading to increased water depths and wave heights at the toe of the structure. This finding allows to quantify and to explain the variability of wave overtopping in storms following one another at intervals shorter than the recovery time of the foreshore.

The authors regret that there was a typo in Equation (14b). 

Ongoing climate change is a significant threat to coastal communities. To understand potential risks during extreme storm events, detailed post-overtopping processes are investigated using DualSPHysics and SWASH with a newly developed approach. It is a calibrated-based wave generation: a target incident wave is first obtained from the validated SWASH model, and DualSPHysics creates the target incident wave by adjusting the offshore wave and bathymetry conditions. This one-way coupling process makes the DualSPHysics computation efficient enough to apply 3D simulation. With a vertical wall at the end of a room located at the end of the promenade in a mild and shallow foreshore, the present model shows a good correspondence on the wave force with the literature. After confirming the efficiency and accuracy of the present model, the 3D simulation with furniture inside the room was conducted and visualized with the state-of-the-art visualization technique. Based on the visualization, the potential risks during the extreme storm event are further discussed in this paper. The present work shows a further capability of DualSPHysics to deal with wave–object–structure interaction based on the latest developments in an efficient way. The developed model can be further used to understand the potential risks of ongoing climate change.

Mangrove vegetation constitutes a natural coastal defence against waves and erosion. Despite their protective role, mangrove ecosystems have experienced continuous degradation over the last decades due to human causes. At retreating mangrove coastlines, bamboo structures are built to create new habitat for mangrove colonization. Existing structures have experienced mixed rates of success due to the lack of a scientific basis in their design. Optimizing future structure designs requires investigating the effect of the bamboo poles on waves. We consequently conducted laboratory experiments to measure wave transformation, hydrodynamic forces, and flow velocities inside cylinder arrays, mimicking bamboo poles, with varying cylinder configurations and orientations. The experiments provided relationships for wave transmission, wave reflection, and the drag coefficients for configurations with volumetric porosities between n = 0.64 − 0.9. Configurations with a small lateral spacing (causing higher blockage) and a relatively longer streamwise spacing (causing less sheltering) exhibit larger forces and dissipation per element. Such arrangements enable optimizing wave dissipation at locations where the wave direction has low variability over the year. Placing the poles horizontally instead of vertically increases the forces and wave dissipation per element in relatively deeper water. Based on the experiments, we developed a conceptual analytical model that predicts wave reflection and dissipation through cylinder arrays, including blockage and sheltering. The model can reproduce the influence of cylinder arrangement on wave transformation, and it suggests that accurate predictions of sheltering and wave reflection are important to find optimal designs. Overall, these results provide useful insights on how to model and optimize the design of structures for mangrove restoration.

The state-of-The-Art formulas for mean wave overtopping (q) assessment typically require wave conditions at the toe of the structure as input. However, for structures built either on land or in very shallow water, obtaining accurate estimates of wave height and period at the structure toe often proves difficult and requires the use of either physical modeling or high-resolution numerical wave models. Here, we follow Goda's method to establish an accurate prediction methodology for both vertical and sloping structures based entirely on deep-water characteristics-where the influence of the foreshore is captured by directly incorporating the foreshore slope and the relative water depth at the structure toe (htoe/Hm0,deep). Findings show that q decreases exponentially with htoe/Hm0,deep due to the decrease of the incident wave energy; however, the rate of reduction in q decreases for structures built on land or in extremely shallow water (htoe/Hm0,deep ≤ 0.1) due to the increased influence of wave-induced setup and infragravity waves-which act as long-period fluctuations in mean water level-generated by nonlinear wave transformation over the foreshore.

The weakly reflective wave generation is a wave generation and absorption method in phase-resolving models, based on the assumption that the waves propagating towards the wave generation boundary are small amplitude shallow water waves with direction perpendicular to the boundary. This assumption makes the method weakly reflective for dispersive and directional waves. The internal wave generation method was proposed by Vasarmidis et al. (2019b) as an alternative, for the non-hydrostatic wave model, SWASH, to avoid reflections. In this study, a comparison is made between the performance of the new internal wave generation method and the weakly reflective wave generation method. It is shown that using the internal wave generation leads to a significantly more accurate prediction of the resulting wave field in case of waves reflected back to the numerical boundary. Additionally, the internal wave generation method is extended to short-crested waves and SWASH is validated for the first time with experimental data for the cases of wave propagation over a shoal and wave diffraction around a wall. The proposed extended internal wave generation method increases the capability of SWASH towards the study of wave propagation of highly dispersive short-crested waves in coastal environments with minimal reflection from the boundaries.

Coastal vegetation has been increasingly recognized as an effective buffer against wind waves. Recent laboratory studies have considered realistic vegetation traits and hydrodynamic conditions, which advanced our understanding of the wave dissipation process in vegetation (WDV) in field conditions. In intertidal environments, waves commonly propagate into vegetation fields with underlying tidal currents, which may alter the WDV process. A number of experiments addressed WDV with following currents, but relatively few experiments have been conducted to assess WDV with opposing currents. Additionally, while the vegetation drag coefficient is a key factor influencing WDV, it is rarely reported for combined wave-current flows. Relevant WDV and drag coefficient data are not openly available for theory or model development. This paper reports a unique dataset of two flume experiments. Both experiments use stiff rods to mimic mangrove canopies. The first experiment assessed WDV and drag coefficients with and without following currents, whereas the second experiment included complementary tests with opposing currents. These two experiments included 668 tests covering various settings of water depth, wave height, wave period, current velocity and vegetation density. A variety of data, including wave height, drag coefficient, in-canopy velocity and acting force on mimic vegetation stem, are recorded. This dataset is expected to assist future theoretical advancement on WDV, which may ultimately lead to a more accurate prediction of wave dissipation capacity of natural coastal wetlands. The dataset is available from figshare with clear instructions for reuse (10.6084/m9.figshare.13026530.v2, Hu et al., 2020). The current dataset will expand with additional WDV data from ongoing and planned observation in natural mangrove wetlands.

Practitioners often employ diverse, though not always thoroughly validated, numerical models to directly or indirectly estimate wave overtopping (q) at sloping structures. These models, broadly classified as either phase-resolving or phase-averaged, each have strengths and limitations owing to the physical schematization of processes within them. Models which resolve the vertical flow structure or the full wave spectrum (i.e. sea-swell (SS) and infragravity (IG) waves) are considered more accurate, but more computationally demanding than those with approximations. Here, we assess the speed-accuracy trade-off of six well-known models for estimating q, under shallow foreshore conditions. The results demonstrate that: i) q is underestimated by an order of magnitude when IG waves are neglected; ii) using more computationally-demanding models does not guarantee improved accuracy; and iii) with empirical corrections to incorporate IG waves, phase-averaged models like SWAN can perform on par, if not better than, phase-resolving models but with far less computational effort.

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 comprised 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 and pattern 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. A detailed analysis of the complex processes preceding the impacts on the vertical wall proves that a correct reproduction of the horizontal impact force and pressures is highly dependent on the accuracy of reproducing the bore interactions.

The vegetation capacity to protect the coasts from wave action is becoming more important and attractive due to ongoing sea level rise and increasing storminess. In addition, it is a quite environmentally friendly way. Quantifying the vegetation effect in wave propagation will be relevant for coastal management. A non-hydrostatic wave model based on the nonlinear shallow water equations, SWASH, offers opportunities to quantify the wave dissipation effect in vegetation fields. However, limited applications of SWASH addressing this subject can be found in the literature and therefore it is important to enhance the existing knowledge on the model behaviour. In this research, in order to understand the characteristics of the SWASH model further, the model is applied to reproduce the significant wave height (H_s) evolution over vegetation fields measured in flume experiments and in field campaign. Overall, SWASH performed very well in reproducing the H_s evolution measured both in the laboratory and in the field. In the case of flume data, the statistical scores MBE, RMSE and MRE, showed that the SWASH performance clearly improved when increasing the number of vertical layers assumed in the simulations. In the case of field data, considering a vegetation factor (V_f ) between 0.1 and 0.5, that represents the overall effect of scarcely known numerical vegetation parameters, led to a fairly good SWASH performance in modelling the H_s evolution over vegetation.

Design criteria for coastal defenses exposed to wave overtopping are usually assessed by mean overtopping discharges and maximum individual overtopping volumes. However, it is often difficult to give clear and precise limits of tolerable overtopping for all kinds of layouts. A few studies analyzed the relationship between wave overtopping flows and hazard levels for people on sea dikes, confirming that one single value of admissible mean discharge or individual overtopping volume is not a sufficient indicator of the hazard, but detailed characterization of flow velocities and depths is required. This work presents the results of an experimental campaign aiming at analyzing the validity of the safety limits and design criteria for overtopping discharge applied to an urbanized stretch of the Catalan coast, exposed to significant overtopping events every stormy season. The work compares different safety criteria for pedestrians. The results prove that the safety of pedestrians on a sea dike can be still guaranteed, even for overtopping volumes larger than 1,000 L/m. Sea storms characterized by deep-water wave height between 3.6 and 4.5 m lead to overtopping flow depth values larger than 1 m and flow velocities up to 20 m/s. However, pedestrian hazard is proved to be linked to the combination of overtopping flow velocity and flow depth rather than to single maximum values of one of these parameters. The use of stability curves to assess people's stability under overtopping waves is therefore advised.

Physical model experiments were conducted in a wave tank at Flanders Hydraulics Research, Antwerp, Belgium, to characterize the wave overtopping and impact force on vertical quay walls and sloping sea dike (1:2.5) under very oblique wave attack (angle between 45° and 80°). This study was triggered by the scarce scientific literature on the overtopping and force reduction due to very oblique waves since large reduction is expected for both when compared with the perpendicular wave attack. The study aimed to compare the results from the experimental tests with formulas derived from previous experiments and applicable to a Belgian harbor generic case. The influence of storm return walls and crest berm width on top of the dikes has been analyzed in combination with the wave obliqueness. The results indicate significant reduction of the overtopping due to very oblique waves and new reduction coefficients were proposed. When compared with formulas from previous studies the proposed coefficients indicate the best fit for the overtopping reduction. Position of the storm return wall respect to the quay edge rather than its height was found to be more important for preventing wave induced overtopping. The force reduction is up to approximately 50% for the oblique waves with respect to the perpendicular wave impact and reduction coefficients were proposed for two different configurations a sea dike and vertical quay wall, respectively.

The work highlights the importance of directional spreading effects on wave overtopping estimation in shallow and mild sloping foreshores. Wave short-crestedness leads, in general, to a reduction of mean overtopping discharges on coastal structures. In the present work, the case of a sea dike with gentle foreshore in very and extremely shallow water conditions is analysed. Physical model tests have been carried out in order to investigate the effect of directional spreading on overtopping and incident wave characteristics. In the present experimental campaign, the effect of wave spreading has only been investigated for perpendicular wave attack. Results show that directional spreading is proved to cause a reduction of average discharge of sea dikes with gentle and shallow foreshore. Expressions for the reduction factor for directional spreading are derived, fitted on the tested database. The use of this reduction factor leads to more accurate prediction and avoids overtopping overestimation, however reduction-factor formulations are overtopping-formula depending.