The overtopping flow velocity and layer thickness are closely related to the stability of coastal structures. Some empirical formulas are available for estimating the overtopping flow velocities and layer thicknesses. However, these empirical equations were derived based on experiments where only limited amount of wave conditions and dike configurations (mostly smooth straight waterside slopes) were tested. Therefore, the extrapolation of existing empirical equations to cases that are outside of the applicable ranges remains uncertain. Numerical modelling has become an important complementary tool to physical experiments. In this study, we developed a 2DV numerical model based on the OpenFOAM framework to simulate the overtopping flow velocity and layer thickness at the waterside edge of the dike crest. This model is validated by comparing the modelled overtopping flow parameters with the measured ones from physical experiments. The model validation shows that the 2DV OpenFOAM model is capable of predicting the overtopping flow parameters with a low probability (2%) of exceedance reasonably well although the overtopping layer thickness is slightly overestimated.

Accurate calculation of wave overtopping is important for determining the required crest height and geometry of a dike. Berms and roughness elements are widely used to reduce the average overtopping discharge at dikes while the reductive effects of berm and roughness are still not fully understood. Several empirical formulae are available to predict the overtopping rate at coastal structures. However, the extrapolation of these empirical formulae is not always applicable for complex structures (e.g. a dike that has a berm and/or roughness elements on the waterside slopes) or wave conditions that are outside the applicability of the empirical predictors. A 2D numerical model based on OpenFOAM® is set up in this study for predicting wave overtopping at dikes that have complex configurations with berms and roughness elements. The validation results show that this OpenFOAM® model is capable of reproducing the incident waves accurately and predicting the wave overtopping discharge with good accuracy. Subsequently, the numerical model is applied to study the reductive influence of a berm and protruding blocks on the mean overtopping discharge at dikes. The roughness of protruding blocks is incorporated by explicitly modelling the protrusions using refined mesh. The model shows reasonable behaviour of the reduction of wave overtopping influenced by a berm and roughness. This indicates the capabilities of the numerical model in the design and safety assessment of dikes.

A reliable estimation of average wave overtopping discharge is important for dike design and safety assessment. Berms and roughness elements are widely applied to reduce the overtopping discharge over dikes. In this study, the effects of a berm and roughness on the wave overtopping discharge are investigated by means of physical model tests. New empirical formulae are derived from the analysis of the experimental data in order to provide more accurate estimations of reductive influence of berms and roughness on the wave overtopping discharge. Additionally, a new formula is developed to estimate the reductive influence of varying roughness along the waterside slopes with a berm. The new equations show a significantly better performance within the tested range when compared with existing formulae for the average overtopping discharge.

The average overtopping discharge is an important parameter for the design and reinforcement of dikes. Rock armour on the waterside slopes and berms of dikes is widely used to reduce the wave overtopping discharge by introducing slope roughness and dissipation of energy in the permeable armour layer. However, methods for estimating the influence of a rock berm and roughness of rock armour at dikes on the average overtopping discharge still need to be developed and/or validated. Therefore, this study aims to develop empirical equations to quantify the reductive influence of rock armour on wave overtopping at dikes. Empirical equations for estimating the effects of rock berms and roughness are derived based on the analysis of experimental data from new physical model tests. The influence of roughness of the rock armour applied on parts of waterside slopes is estimated by introducing the location weighting coefficients. Results show that the newly derived equations to predict the average overtopping discharge at dikes lead to a significantly better performance within the tested ranges compared to existing empirical equations.

The average overtopping discharge is an important parameter for the design of flood defences. Several empirical formulas are available for predicting the overtopping discharge at dikes. However, these empirical formulas often have their specific applicable conditions. To complement with the empirical methods, a numerical model has been developed using the open source CFD package OpenFOAM to model the wave overtopping at dikes. Systematic calibration and validation of the numerical model are performed. The influences of the mesh, solver, turbulence model and roughness height on the modelled results of the average overtopping discharge have been investigated during the model calibration. The simulations show that the turbulence model increases the accuracy of the numerical model for predicting the average overtopping discharge under wave breaking conditions. The calibrated model is then validated by comparing the modelled average overtopping discharges with the measured ones from the physical model tests. Results show that the OpenFOAM model is capable of predicting the average overtopping discharge accurately at dikes that have a smooth straight waterside slope.

To guarantee port accessibility, navigation approach channels need to be well maintained. Annual dredging efforts to maintain navigable channels may well exceed tens of millions m³ of sediment per year, which results in high recurrent costs for port operators. Quantification of expected siltation rates with process-based numerical models helps to effectively design and optimise approach channels. The setup of such models requires several assumptions and parameter settings which introduce uncertainty in model output. However, traditional Monte Carlo methods to quantify that uncertainty in model output are often too resource-intensive with current standard computer resources to be feasibly applied in coastal engineering projects such as approach channel design. Here, we use an alternative multifidelity approach to estimate the probability density function of channel siltation, at lower computational costs compared to direct Monte Carlo simulation. The idea behind this method is to map the output uncertainty of a faster, but inaccurate model to a preferred high-detailed model. The key requirement is that the faster, low-fidelity model and the detailed high-fidelity model are correlated, and that his correlation can be modelled with a probabilistic function. Since linearity of the correlation is not a requirement, the coarse-grid model can be very inaccurate but still serve as an adequate predictor of the high-fidelity model. In this study we did observe a highly nonlinear correlation, which in our case is explained by underestimation of channel siltation near the surf zone by the coarse model. In the presented multifidelity approach we adopted a combination of quasi-random Monte Carlo simulation and a non-parametric Gaussian process transfer function to estimate the uncertainty of total siltation and spatial patterns of siltation in a port approach channel. We argue that the multifidelity approach is conceptually straightforward and found that it can be used to significantly decrease the costs of probabilistic analysis; in our case we found a 85% decrease compared to direct Monte Carlo simulation. An additional advantage is that the approach allows for a trade-off between precision and efficiency by varying the number of high-fidelity model runs. Therefore, we conclude that the multifidelity framework is a potential powerful alternative for cases in which direct Monte Carlo simulation is infeasible or undesirable.

An accurate description of the maximum flow velocity across flood-protective structures is necessary in order to determine the stability of the landward slope and the amount of cover erosion. In this paper, two new formulas are derived to describe the change in the maximum overtopping flow velocity along the dike crest and the adjacent landward slope. These formulas are coupled in an effort to accurately predict the velocities in wave overtopping events. This analytical model is validated using 244 data points from flume tests and 300 data points from field tests on river dikes in the Netherlands. The modelled flow velocity shows good agreement with the measured flow velocity, with a Nash-Sutcliffe efficiency factor varying from 0.49 to 0.87. Also, the derived formulas are compared with existing formulas for wave overtopping flow velocities and overall showed a better performance for a wide range of geometries and covers.

Transitions in the dike revetment or in the grass cover can significantly affect the wave overtopping discharge and the dike cover erosion. At the University of Twente, two PhD students recently started on the challenge of quantifying the effect of (1) waterside transition on the wave overtopping discharge and (2) transitions in grass covered dikes on dike erosion. In this paper we present their preliminary results and outline their future plans. Firstly, new laboratory experiments show that the existing wave overtopping formulas are not able to accurately predict the overtopping discharge in case of transitions on the waterside slope. Secondly, the analytical dike cover erosion model shows that transitions in grass covers significantly affect the location of maximum flow velocity and potential dike cover erosion. In future work, detailed numerical models will be developed for both the waterside slope and the landward slope to further increase our understanding of the effects of transitions on the wave overtopping discharge and the dike cover erosion.

Structures integrated in a grass-covered dike may increase erosion development. Currently, safety assessment methods for flood defences are only applicable for a conventional grass-covered dike and the effects of structures on dike cover erosion are poorly understood. Since many dikes have a road on top, it is important to study the effect of such a road structure on erosion onset during wave overtopping. To investigate this effect, a coupled hydrodynamic–eroison model was developed. The erosion onset caused by overtopping waves was predicted by combining the time-varying bed shear stresses from the hydrodynamic model with a depth-dependent erosion model. The results show that roads on top of a dike increase the erosion of the neighbouring grass cover. This increase in erosion may have a negative impact on dike stability. Therefore, we recommend considering effects of constructions on top of dike profiles during safety assessments. Explicitly, consideration of the roughness transitions in the safety assessments of dikes is recommended.

The article was published Open Access under the Dutch Compact Agreement; however, due to an internal system error, previous HTML rendering of the article did not reflect this.

Hard structures, i.e., roads, are commonly found over flood defences, such as dikes, in order to ensure access and connectivity between flood protected areas. Several climate change future scenario studies have concluded that flood defences will be required to withstand more severe storms than the ones used for their original design. Therefore, this paper presents a probabilistic methodology to assess the effect of a road on top of a dike: it gives the failure probability of the grass cover due to wave overtopping over a wide range of design storms. The methodology was developed by building two different dike configurations in computational fluid dynamics Navier-Stokes solution software; one with a road on top and one without a road. Both models were validated with experimental data collected from field-scale experiments. Later, both models were used to produce data sets for training simpler and faster emulators. These emulators were coupled to a simplified erosion model which allowed testing storm scenarios which resulted in local scouring conditioned statistical failure probabilities. From these results it was estimated that the dike with a road has higher probabilities (5 × 10^-5 > P_f > 1 × 10^-4) of failure than a dike without a road (P_f < 1 × 10^-6) if realistic grass quality spatial distributions were assumed. The coupled emulator-erosion model was able to yield realistic probabilities, given all the uncertainties in the modelling process and it seems to be a promising tool for quantifying grass cover erosion failure.

Piping erosion has been proved to be one of the failure mechanisms that contributes the most to the total probability of failure on the Dutch flood defence systems. The present study aimed to find the impact of correlation and tail dependence between soil parameters present in the Sellmeijer revised limit state equation for piping safety assessment, particularly between the grain size and hydraulic conductivity parameters. A copula based random sampling method was used as a tool to include this effect in the probabilistic estimation of this type of failure. The method was framed in a real case study for a flood defence along the Lek river, in The Netherlands. The results showed that inclusion of correlation between the two parameters reduces the variance of the limit state marginal distribution by almost 10% when compared to the uncorrelated case. This effect changes the tail values sampling frequency and therefore reduces the probability of failure by a factor of 1.7. The omission of correlation between the two parameters for safety assessment based on the Sellmeijer limit state function may result in over dimensioned structures.

Piping erosion is one of the major causes of failure of flood defences. The occurrence of this failure mechanism is difficult to predict and it can be triggered during a flood event. The inclusion of hard structures under flood defences will change the probability of occurrence of this erosion process. The present study aimed at understanding the effect of an embedded sewer pipe under the flood defence foundation in its safety assessment for piping erosion failure. This was done by setting a probabilistic analysis framework based on emulation of finite element models of porous media flow and the fictitious permeability approach. The effects of size and location of the sewer pipe were evaluated via a deterministic safety factor approach. Afterwards, emulators of the most safe and unsafe models were trained and used for probabilistic assessments. The results showed that the embedment of a sewer pipe in the flood defence foundation has a significant effect on its safety. The magnitude of the effect is highly dependent on the size and location of the sewer pipe. Furthermore, the foundation permeability uncertainty shows a conditional effect with respect to the sewer pipe size and location.