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%.

This study characterises the flow velocity of individual extreme waves that overtop promenades using the bubble image velocimetry (BIV) technique (Ryu et al., 2005). Experimental tests were carried out in the small-scale wave flume CIEMito, at the Marine Engineering Laboratory (LIM) of the Universitat Politècnica de Catalunya – BarcelonaTech (UPC), and the obtained images were post-processed to calculate the flow velocities. The ultimate objective of the experimental campaign is to develop more precise models for forecasting wave overtopping of structures with an emergent toe, commonly found on sandy beaches and frequently utilized as promenades or waterfronts in most urbanized coastal environments. The NewWave theory (Tromans et al., 1991) was used to simulate the extreme individual wave overtopping in a real random sea state. The NewWave theory establishes a correlation between the expected form of a large wave in a linear sea state and the bulk characteristics of the sea state. Using focused wave groups instead of long-duration irregular wave time series offers several benefits. It improves the ability to repeat experiments and enhances measurement capabilities by providing greater temporal resolution in models used to investigate significant wave interactions (Hofland et al., 2014). Additionally, due to the compactness of focused wave groups, wave absorption becomes unnecessary. The research identifies two distinct case studies with varying wave forcing, tidal regimes and coastal layouts. This work presents a case study of a typical Mediterranean Sea configuration, which is a micro-tidal environment with steep and relatively short foreshores and pocket beaches. The study aims to characterize the overtopping flow velocity on the selected structure, and the feasibility of non-intrusive measurements such as a BIV technique has been investigated. The work includes preliminary results.

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.

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.

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

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.

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.

Wave height attenuation in vegetation canopies is often all attributed to the drag force exerted by vegetation, whereas other potential dissipation process is often neglected. Previous studies without vegetation have found that opposing currents can induce wave breaking and greatly increase dissipation. It is not clear if similar process may also occur in vegetation canopies. We conducted systematic flume experiments to show that wave breaking in opposing currents can occur in vegetated flows, but only in submerged canopies with shear currents above vegetation top. Subsequently, we developed a new analytical model to understand and assess the contribution of both drag-induced dissipation in the lower vegetation layer and current-induced breaking in the upper free layer. A new generic drag coefficient relation was applied in the model to quantify drag-induced dissipation with various current-wave combinations. It shows that breaking induced by opposing currents constitutes an essential part (up to 87%) of the total dissipation, which leads to considerably higher dissipation than the cases with following currents. Breaking can occur with various submergence ratios and with small opposing currents in the submerged vegetation field. It indicates that similar breaking process is likely to occur in real vegetation fields. The present study reveals and quantifies the current-induced wave breaking process that has not been reported before, which can improve our understanding of vegetation wave dissipation capacity in field conditions.

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.

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.

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.

Despite the widely recognized role of infragravity (IG) waves in many often-hazardous nearshore processes, spectral wave models, which exclude IG-wave dynamics, are often used in the design and assessment of coastal dikes. Consequently, the safety of these structures in environments where IG waves dominate remains uncertain. Here, we combine physical and numerical modeling to: (1) assess the influence of various offshore, foreshore, and dike slope conditions on the dominance of IG waves over those at sea and swell (SS) frequencies; and (2) develop a predictive model for the relative magnitude of IG waves, defined as the ratio of the IG-to-SS-wave height at the dike toe. Findings show that higher, directionally narrow-banded incident waves; shallower water depths; milder foreshore slopes; reduced vegetated cover; and milder dike slopes promote IG-wave dominance. In addition, the empirical model derived, which captures the combined effect of the varied environmental parameters, allows practitioners to quickly estimate the significance of IG waves at the coast, and may also be combined with spectral wave models to extend their applicability to areas where IG waves contribute significantly.

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.

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.

Due to ongoing climate change, overtopping risk is increasing. In order to have effective countermeasures, it is useful to understand overtopping processes in details. In this study overtopping flow on a dike with gentle and shallow foreshores are investigated using a non-hydrostatic wave-flow model, SWASH (an acronym of Simulating WAves till SHore). The SWASH model in 2DV (i.e., flume like configuration) is first validated using the data of long crested wave cases with second order wave generation in the physical model test conducted. After that it is used to produce overtopping flow in different wave conditions and bathymetries. The results indicated that the overtopping risk is better characterized by the time dependent h (overtopping flow depth) and u (overtopping flow velocity) instead of h_max (maximum overtopping flow depth) and u_max (maximum overtopping flow velocity), which led to overestimation of the risk. The time dependent u and h are strongly influenced by the dike configuration, namely by the promenade width and the existence of a vertical wall on the promenade: the simulation shows that the vertical wall induces seaward velocity on the dike which might be an extra risk during extreme events.

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.

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.