# Validation of RANS Modelling for Wave Interactions with Sea Dikes on Shallow Foreshores Using a Large-Scale Experimental Dataset

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

^{®}, 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.

## 1. Introduction

^{1}s)) shoal and eventually break, transferring energy to both their super- and subharmonics (or long waves: hereafter LW, O(10

^{2}s)) by nonlinear wave-wave interactions. Further pre-overtopping hydrodynamic processes along the mildly sloping foreshore include wave dissipation by breaking (turbulent bore formation) and bottom friction, reflection against the foreshore and dike, and wave run-up on the dike slope. Finally, waves overtop the dike crest, and post-overtopping processes include bore propagation on the promenade, bore impact on a wall or building, and reflection back towards the sea interacting with incoming bores on the promenade.

^{®}is chosen because of its increasing popularity for application to wave-structure interactions. Validation of this numerical model is done by reproducing large-scale experiments of overtopped wave impacts on coastal dikes with a very shallow foreshore from the WAve LOads on WAlls (WALOWA) project [29]. The large-scale nature of these experiments reduces the scale effects significantly compared to small-scale experiments, which can be particularly important to the wave impacts on the dike-mounted vertical wall, especially in case of plunging breaking bore patterns and impulsive impacts [30].

## 2. Methods

#### 2.1. Large-Scale Laboratory Experiments

^{−2}m)) were noted between the profile measurements before and after [32], a fixed bed is a reasonable assumption for the numerical modelling. In addition, the repeated test makes validation of the numerical model possible relative to the experimental uncertainty.

_{0}= tan α/(H/L

_{0})

^{1/2}with α the foreshore slope angle, H the wave height, and L

_{0}the deep water wave length [34]: 0.5 < ξ

_{0}≈ 0.7 < 3.3) and spilling breakers on the 1:35 foreshore slope (ξ

_{0}≈ 0.2 < 0.5). Considering this was a test of a dike with a very shallow foreshore depth (Table 1: 0.3 < h

_{t}/H

_{m0,o}< 1.0 [35]), the wave energy at the toe of the dike was dominated by LW energy.

_{o}) was measured with resistance-type wave gauges (WG) deployed at seven different locations along the Delta Flume side wall (Figure 1 and Figure 2a). WG02–WG04 were installed over the flat bottom part of the flume close to the wave paddle. These wave gauges were positioned to allow a reflection analysis following the method of Mansard and Funke [36]. WG07 was installed along the transition slope; WG11 and WG13 along the foreshore slope. WG14 was installed close (~0.35 m) to the dike toe. The data of WG11 are not considered further in the present analysis because of faulty data. Furthermore, to remove unwanted noise in the η signals measured by the other WG’s from the wave paddle up to the dike toe, a low-pass 3rd order Butterworth filter with a cut-off frequency of 1.50 Hz was applied. This frequency is well above the frequencies of the super-harmonics of the primary waves and frequency components due to triad interactions between the primary components and the difference frequency, which gain energy in the shoaling and surf zone [37].

_{x}in one direction (i.e., towards the wall) 0.026 m above the promenade. Additionally, a bidirectional electromagnetic current meter (ECM, Figure 2c) was installed at the same cross-shore location as WLDM02 and PW02 to obtain directional information of the incoming or reflected flow. The ECM disc was positioned 0.03 m above the promenade and sampled the horizontal velocity at 16 Hz. Further detailed information on the sensor setup on the promenade and the post-processing of the η and U

_{x}data measured on top of the promenade was provided by Cappietti et al. [38]. During return flow, positive U

_{x}values were possibly incorrectly measured by the PWs, indicated by the ECM that measured negative U

_{x}values during return flow (compared to the measurements of the co-located PW02). This will be further discussed in the comparison with the numerical model result (Section 3.1). However, no such co-located measurements are available for other paddle wheels than PW02, so no correction of the PW measurements during return flows was attempted.

_{x}and pressure p measurement systems integrated into the wall. The horizontal impact force was measured by two compression-type load cells (LC) connecting the same hollow steel profile to the very stiff supporting structure (Figure 2e). Impact pressures were measured by 15 pressure sensors (PS). The first 13 PSs were spaced vertically over a metal plate flush mounted in the middle section of the steel wall, with PS14 and PS15 placed horizontally next to PS05 or the fifth PS from the bottom (Figure 2f). The initial post-processing of the F

_{x}and p signals, including baseline correction and filtering, is discussed by Streicher [39]. Additional filtering was applied to remove the high frequency oscillations caused by stochastic processes during dynamic or impulsive impacts, so that the signal can be reproduced by a deterministic numerical model [40]. To achieve this, an additional 3rd order Butterworth low-pass filter with a cut-off frequency of 6.22 Hz was necessary. This corresponds to a cut-off frequency of 3.0 Hz at a prototype scale, which is still well above the natural frequency of about 1.0 Hz for typical buildings found along, e.g., the Belgian coast [41]. Furthermore, local spatial variability over the width of the flume of the resultant F

_{x}(i.e., derived from the LCs and pressure integrated) and p (i.e., PS05, PS14, and PS15) time series was found to be low (not shown). This spatial variability over the width of the experimental flume was therefore further neglected in the quantitative numerical model validation: for F

_{x,}the LC-derived signal was used and for p, the PS05 signal was used.

#### 2.2. Numerical Model

#### 2.2.1. Model Description

#### 2.2.2. Computational Domain and Mesh

_{o}is divisible by it (i.e., h

_{o}/Δz = 4.14/0.045 = 92), meaning that the SWL can lie perfectly along cell boundaries, or in other words, α-values between 0 and 1 are thereby minimised at the start of the simulation, which simplifies the initialisation of the SWL and is beneficial for an effectively still SWL at the start of the simulation.

_{b}= ~120 m), this is considered an acceptable assumption. Both the refinement level in the refinement zones around the surface elevation zones (β

_{sez}) and the maxCo were verified in a convergence analysis (Appendix A).

#### 2.2.3. Boundary Conditions

^{+}(i.e., 1 < z

^{+}< 300) is complied with. For the remaining boundary conditions, initial conditions, and solver settings, the same settings were chosen as those reported by Devolder et al. [48].

#### 2.2.4. Data Sampling and Processing

_{SW}and η

_{LW}by applying a 3

^{rd}-order Butterworth high- and low-pass filter, respectively. A separation frequency of 0.09 Hz was employed, which is in between the bound long wave frequency (f

_{1}–f

_{2}= 0.035 Hz) and the lowest frequency of the primary wave components (f

_{2}= 0.155 Hz).

#### 2.3. Validation Method

_{r}[74]:

_{r}; MAE is the mean absolute error defined by:

_{r}is bounded by [−1.0, 1.0] and, in general, more rationally related to model accuracy than other existing model performance indices or skill scores. For the purposes of this paper, d

_{r}is used as a general measure of the model performance, and a d

_{r}value of 0.5 is already considered to be a poor model performance. Since it is a single measure of model performance, it can be more easily used to evaluate, for example, the spatial model performance over the length of the wave flume.

_{r}can be evaluated between REXP and EXP as well. This can serve as a limit above which a d

_{r}value of the numerical model signifies that the numerical model performance cannot be improved beyond the experimental model uncertainty due to model effects, etc. Therefore, similar to the relative errors as defined by van Rijn et al. [75], a relative refined index of agreement d′

_{r}is proposed here which provides the performance of the numerical model relative to the experimental model uncertainty:

_{num}–MAE

_{rexp}is negative (i.e., <0), the numerical error compared to the experiment is smaller than the experimental uncertainty, which means that the numerical model performance cannot be improved. In that case MAE

_{num}–MAE

_{rexp}= 0 is forced, so that d′

_{r}= 1. A classification of model performance based on ranges of d′

_{r}values and corresponding rating terminology is proposed in Table 2.

_{p}and σ

_{o}are the standard deviations of the predicted and observed time series, respectively. Another important statistical parameter is the bias B, given by:

- unnecessarily penalising the numerical model performance for an experimental measurement error. For example, in the experimentally measured and processed time series of p and F
_{x}, often some residual instrumental noise or oscillations persisted during such non-event (or “dry”) times; - unnecessarily rewarding the model performance towards (almost) perfect agreement. For example, during the time between impacts no water reaches the wall and model performance would be perfect during such times (disregarding measurement noise).

## 3. Results

#### 3.1. Time Series

_{x}on the promenade in Figure 5, and the total horizontal force F

_{x}and pressures p on the vertical wall in Figure 6 and Figure 7, respectively.

_{x}on top of the promenade appears to significantly underestimate the experimental measurements (Figure 5). This underestimation mostly disappears when using the OF depth-averaged velocity ${\overline{U}}_{x}$ instead, which is done for the remainder of the validation. In addition, OF shows much better correspondence to the ECM than the PWs during return flow of a reflected bore (U

_{x}< 0). This confirms that the PWs did not measure correct velocities during those instances (e.g., 57 s ≤ t ≤ 63 s in Figure 5b–c).

_{x}and p on the vertical wall, OF generally reproduces the timing of the impact events, including the evolution over time (Figure 6 and Figure 7). However, the EXP time series peak values appear to be underestimated by OF for both F

_{x}and p, and for a few impacts, the first dynamic impact peak is not entirely captured either (e.g., t = 82 s and 140 s). In the experiment, the lowest PSs were loaded more often than the PSs positioned higher up the vertical wall, because of different bore impact run-up heights. The lowest PSs also registered the highest values, indicating a mostly hydrostatic pressure distribution along the vertical wall [76]. Both these observations were reproduced by OF. Validation of the pressure distribution along the vertical wall is further investigated in Section 3.4.

#### 3.2. Wave Characteristics

_{rms}is calculated in the time domain and represents a characteristic wave height and measure of the wave energy. The evolution of H

_{rms}, the short- and long-wave components (i.e., H

_{rms,sw}and H

_{rms,lw}), and the mean surface elevation $\overline{\eta}$ or wave setup over the wave flume up to the toe of the dike are displayed in Figure 8. The experimental repeatability of H

_{rms}appears to be near-perfect, since the EXP and REXP data points are almost indistinguishable. The OF results for these wave characteristics are available along the complete distance from the wave paddle until the toe of the dike location. The numerical results seem to follow the experiments very well, although some discrepancies can be seen. The total and SW wave heights (H

_{rms}and H

_{rms,sw,}respectively, in Figure 8) decrease in the OF result from the wave paddle up to the toe of the foreshore and underestimate the EXP wave height along this distance. Over the foreshore, the SWs start to shoal until their steepness becomes too high and, according to OF, start to break about 11 m from WG07 towards the dike. The location of incipient wave breaking (or decrease in H

_{rms}), x

_{b}, cannot be validated with the experiment, because of insufficient wave gauges in the wave breaking zone. In any case, the EXP wave height increase due to shoaling (WG07) and decrease due to breaking (WG13–14) are reproduced well by OF. However, over the foreshore, OF slightly underestimates the wave amplitude. The experimental LW wave height (H

_{rms,lw}in Figure 8) is slightly underestimated by OF in front of the wave paddle (WG02–WG04), and at the dike toe (WG14).

_{OF}remains close to zero). Further along the flume in the surf zone, however, $\overline{\eta}$ is better predicted by OF, showing a smaller overestimation.

#### 3.3. Model Performance and Pattern Statistics

_{r}and R at the WG locations along the wave flume up to the toe of the dike is visualised for η

_{SW}(d

_{r,sw}and R

_{sw}), η

_{LW}(d

_{r,lw}and R

_{lw}), and η (d

_{r,tot}and R) in Figure 9 and Figure 10 respectively, and of d

_{r}for η and U

_{x}on the promenade in Figure 11.

_{r,tot}along the flume is very similar for both REXP and OF (Figure 9 and Table 3): it remains constant until the shoaling zone (WG02–WG07), decreases over the surf zone (WG07–13), and increases back up to the dike toe (WG13–14). This indicates that the decreased experimental model repeatability of the surface elevation in the surf zone is at least part of the cause of the decreased numerical model performance. The relative model performance d′

_{r}for η is consequently fairly constant, corresponding to a model performance rating of Very Good, which remains consistently so up to the last sensor location in front of the vertical wall. Considering η

_{SW}and η

_{LW}separately reveals that d

_{r,sw}mostly follows the same trend as d

_{r,tot}, and that d

_{r,lw,OF}clearly has a different behaviour: d

_{r},

_{lw,OF}is not as high as d

_{r},

_{sw,OF}in front of the wave paddle (i.e., d

_{r},

_{lw,OF}= ~0.70 and d

_{r},

_{sw,OF}= ~0.85 at WG02–WG04), but steadily increases towards the dike toe, while d

_{r},

_{lw,rexp}remains relatively constant, causing d′

_{r}to slightly increase as well.

_{sw}has a higher weight in R there. Conversely, the dike toe (WG14) is located at an antinode, and therefore R

_{lw}has higher weight in R than R

_{sw}, leading to an increase of R again at the dike toe.

_{r}for η and U

_{x}is shown in Figure 11 and at first sight seems to indicate that the OF model performance for U

_{x}is much worse than that for η, primarily for comparisons with the PW measurements, but also for the ECM measurement. Taking into account the experimental uncertainty, however, the model performance rating for U

_{x}of ECM is actually Very Good (d′

_{r,ECM}in Table 4), which is the same as the OF model performance rating for η on the promenade (d′

_{r,WLDM01-04}in Table 3). For the PW measurements, the OF rating for U

_{x}is still worse (Reasonable/Fair to Bad), but was explained before by the fact that the PW’s had faulty positive U

_{x}measurements during return flow (Section 3.1).

_{x}(B* < 0). Conversely, the bore wave height is well-represented on the promenade (${\sigma}_{WLDM01-04}^{\ast}$ = ~1.00), while the wave height is underestimated at the dike toe (${\sigma}_{WG14}^{\ast}$ = 0.89). The surface elevation phase difference between OF and EXP observed at the dike toe (R

_{WG14}= 0.91) is carried over on the promenade (R

_{WLDM01–04}= ~0.90), but higher phase differences are detected for U

_{x}(R

_{ECM}= 0.73).

_{x}is evaluated at the vertical wall (Figure 12 and Table 5). Both REXP and OF show the highest model performance at the lowest pressure sensor location and a more or less linear decreasing model performance at PS locations higher along the vertical wall. The relative difference between the d

_{r}of REXP and OF increases more along the vertical wall, leading to a numerical model performance rating from Very Good for PS01–PS06, to Good for PS05–PS11, and finally to Reasonable/Fair at the highest PS locations (PS12–PS13) (Table 5). Considering that the bottom PSs registered the highest p values and are therefore the most determinative in the calculation of F

_{x}, it follows that the numerical model performance for F

_{x}is rated Very Good as well. The pattern statistics in Table 5 reveal the remaining numerical errors to be that p and F

_{x}are generally underestimated by OF (i.e., B* < 0.00 and σ* <1.00) and that the impact events still slightly mismatch in time between OF and EXP (R < 1.00).

#### 3.4. Bore Interactions and Impact

_{x}time series or the F

_{x}time series (e.g., peaks, troughs, …), making sure a relevant comparison is made of the bore interaction and the velocity or pressure profile.

_{x,1}(Figure 13e), and ran vertically up the wall temporarily reducing F

_{x}during maximum run-up (not shown). The following run-down and reflection from the wall correspond to a second force peak F

_{x,2}, this time of quasi-static nature (Figure 13f). This type of bore interaction was called a “plunging breaking bore pattern” by Streicher et al. [75], which (in this case) caused a quasi-static impact (F

_{x,1}/F

_{x,2}< 1.20, according to Streicher et al. [75]). This is valid for both the experiment and the numerical model result, indicating that OF was able to reproduce these processes leading to a very similar shape of the pressure distribution along the vertical wall (see pressure profiles in Figure 13d–f) and time evolution of F

_{x}(see time series graph insets in Figure 13d–f). Comparing U

_{x,ECM}from EXP with the velocity profile from OF at the ECM location (see velocity profiles in Figure 13a–c) reveals that OF locally, but consistently underestimated U

_{x}at the vertical measurement position of the ECM, which was also observed in Figure 5b.

_{x}peak, p is severely underestimated, but the distribution is still similar, with a local peak at PS04. The p-profiles differentiate more at the F

_{x}peak of the OF result (Figure 14d) and at the quasi-static F

_{x}peak in the EXP result (Figure 14e). In the experiment, a quasi-hydrostatic pressure profile was measured, at both those time instants. In the OF result, however, a pressure peak is found at PS06, caused by a vortex formed at the foot of the vertical wall upon which a strong flow impinged on the wall at that location. After reflection of the bore, both models correspond again, showing a hydrostatic pressure profile along the wall (Figure 14f).

## 4. Discussion

#### 4.1. Wave Transformation Processes Until the Dike Toe

_{rms}. This section discusses the processes related to the LW transformations over the foreshore as modelled by OF and their correspondence to observations in EXP.

_{m}of the SWs is high for the considered bichromatic wave conditions (Table 1), indicating that the incident-bound LW amplitude was relatively high as well. Furthermore, the normalised bed slope parameter β

_{b}can be calculated [37]:

_{x}is the foreshore slope (= 1:35), ω is the radial frequency of the bound LW (= 2π(f

_{1}–f

_{2})), g the gravitational acceleration, and h

_{b}a characteristic breaking depth (= 2.12 m at x

_{b}= 115 m). A value of 0.28 is obtained, which means that the bound LW shoaling had a mild slope regime (β

_{b}< 0.3), so that the growth rate of the incoming LWs was much higher than that given by Green’s Law (conservative shoaling), indicating significant energy transfer from the primary SWs to the bound LW [77]. Additionally, in a mild-slope regime, LW shoreline dissipation and shoreline reflection are high and low, respectively [37]. However, the beach considered here is not a beach by itself, but acts as a foreshore to a steep-sloped dike. Consequently, no such expected decrease in LW energy towards the shoreline is observed (i.e., H

_{rms,lw}in Figure 8). Indeed, the dike was positioned in the shoaling zone of the long waves, thereby preventing the LWs from breaking. Instead, LWs reflected against the dike, indicated by the oscillations of H

_{rms,lw}towards the dike in the OF result, which implies the presence of a (partial) standing wave system. Wave gauges WG13 in the inner surf zone and WG14 at the dike toe were positioned at a node and anti-node of this standing wave system. This is also clearly visible in the η time series plot, where η

_{LW}is much closer to zero at WG13 (Figure 4e) than at WG14 (Figure 4f). In the surf zone the LW previously bound to the wave group became a free wave, traveling at its own wave celerity. Due to first-order wave generation at the boundary, other spurious free LWs were generated as well at the wavemaker and propagated as free waves towards the dike [78]. During a standing LW crest at the dike toe, the LWs themselves overtopped the dike (i.e., when η > freeboard R

_{c}= 0.117 m, Figure 4f) thereby temporarily aiding several breaking SWs to overtop the crest of the dike (the wave length of the free LWs was more than five times longer than the primary SW components in the inner surf zone). These results have illustrated OF’s ability to reproduce the wave energy transfer to the subharmonics and LW transformations over the foreshore until the dike toe. All these observations also confirm that the contribution of LWs to the processes on the dike, including the wave impact loading on the vertical wall, is very important in the case that is considered here.

#### 4.2. Importance of Differences in Wave Generation Methods

_{x}and η at the boundary, to overcome a possible underestimation of the incident wave height. Such a calibration of the OF model (with a tuning factor of 1.13) was found to solve the underestimation of the wave height (not shown), but introduced or exacerbated other errors, finally leading to lower values of d

_{r}and decreased model performance ratings for U

_{x,ECM}and F

_{x}.

_{rms}(and consequently $\overline{\eta}$) increased in the surf zone, exacerbating the $\overline{\eta}$ overestimation there (not shown). The root cause of this difference is likewise related to the different wave generation methods applied in EXP and OF. In the experimental wave flume, the finite body of water and conservation of mass caused water mass to be redistributed from offshore to the surf zone during build-up of the wave setup, thereby causing a lowering of the mean water level in the offshore region. This process developed differently in OF because of the static boundary condition including AWA. The AWA assures a constant mean water level at the boundary [8,53], meaning that a net water mass is added to the computational domain until a quasi-steady state is achieved when wave setup is fully developed [55]. In this case, OF’s method is closer to the field condition, where generally a large enough body of water is available to supply water mass for the wave setup to develop without noticeably lowering the offshore mean water level. Nevertheless, in the context of the validation, this difference in $\overline{\eta}$ is the cause of many of the remaining inaccuracies in the OF result compared to EXP, because the waves propagated in slightly different mean water depths, which affected the non-linear wave-wave interactions and wave phases in the surf zone. Consequently, it is believed to be the root cause of the strong decrease of R

_{sw}observed in the surf zone (i.e., locations WG13–14 in Figure 10).

_{rms}and overestimation of $\overline{\eta}$) are both attributable to the differences in wave generation methods applied. Although an overall Very Good model performance rating was achieved by OF, it is expected that even better results can be obtained by applying a closed dynamic wave boundary condition in OF, which mimics the EXP wave paddle movement. However, application of the dynamic boundary condition of olaFlow proved to be highly unstable for the present case, and no result was achieved to confirm this hypothesis.

#### 4.3. OF Model Performance for Impacts on a Dike-Mounted Vertical Wall

_{x}at the vertical wall can be explained by a generally correct reproduction of bore interactions over the promenade of the dike. Conversely, discrepancies (even small ones) in bore interactions between OF and EXP can lead to significant differences in the impact type on the vertical wall, and consequently in p and F

_{x}(Section 3.4). In addition, the much lower values of ${B}_{OF}^{\ast}$ and R

_{OF}compared to ${B}_{REXP}^{\ast}$ and R

_{REXP}for U

_{x,ECM}(i.e., ${B}_{REXP}^{\ast}$ = −0.02 and R

_{REXP}= 0.87, ${B}_{OF}^{\ast}$ = −0.25 and R

_{OF}= 0.73 in Table 4) indicate an important contribution of the underestimation of U

_{x}and of phase differences in U

_{x}between OF and EXP to the remaining errors in the impact prediction by OF. The bore interactions on their part depend on the wave conditions at the dike toe location. This is illustrated by the calibrated OF model results, which were found to improve the wave height reproduction at the dike toe compared to the OF model (Section 4.2), while errors increased for the wave setup and wave phases at the dike toe location, leading to a lower model performance for the processes on the dike (not shown).

- 3D effects in EXP (i.e., irregular and oblique wave fronts, wave breaking-induced 3D vortex formation), which are unreproducible by a 2DV RANS model;
- Water-air mixing in bores and air pressure fluctuations in entrained air pockets by overturning wave impacts on the wall, which are both processes not resolved by a multiphase numerical model of two incompressible and immiscible fluids.
- The applied VOF method, which is known to smear the water-air interface over several grid cells and to cause high spurious velocities in the air phase [45]. These limitations may be (partially) overcome by applying the following recent developments:
- o
- An alternative geometric VOF method, isoAdvector, has been developed to obtain a sharper interface [79,80], specifically with applications for marine science and engineering in mind [46]. However, the sharper interface may lead to a larger error in the velocity near the water surface [45] and the method is currently mainly tested and validated for wave propagation, but not yet for wave breaking, overtopping, and wave impact, all processes essential for this study.
- o
- Spurious velocities may be avoided by implementing, e.g., the ghost fluid method [81], although such an implementation is currently not available in any of the open source OpenFOAM versions.

- The turbulence model, which has been carefully chosen as the state-of-the-art (Section 2.2.1), but is still limited by its inherent assumptions.
- Douglas and Nistor [82] have shown that (compared to a dry-bed condition) a bore propagating over a thin layer of water on the bed (i.e., wet-bed condition) can substantially increase the steepness and depth of the bore-front and consequently affect the impact of the bore on the wall. The near-bed resolution of the OF grid along the promenade might not have been able to correctly reproduce wet-bed bore propagation in cases of a very thin layer of water, possibly even modelling a dry-bed bore propagation instead.
- Differences between OF and EXP in the treatment of friction on the bed of the promenade. The no-slip boundary condition and applied wall function in OF modelled a boundary layer, which lowered U
_{x}close to the bed more than was measured in EXP. On average, U_{x}was underestimated by OF at the measurement locations of the PWs and ECM close to the promenade bed (Figure 5, B* in Table 4 and Figure 13a–c).

## 5. Conclusions

^{®}with olaFlow wave boundary conditions (OF), was applied in 2DV for bichromatic wave transformations over a cross-section of a hybrid beach-dike coastal defence system, consisting of a steep-sloped dike with a mildly-sloped and very shallow foreshore, and finally wave impact on a vertical wall. OF was not validated before in this context, where (prior to impact) waves undergo many nonlinear transformations and interact with a dike slope and promenade. A large-scale experiment of bichromatic waves and its repetition were selected for this validation. The repeated test allowed us to assess the accuracy of the measurements, uncertainty due to model effects, and variability due to stochastic processes in the experiment.

_{r}, calculated for OF and the repeated test REXP with reference to the first test EXP, a relative refined index of agreement d

_{r}′ was proposed, which takes the experimental uncertainty, derived from REXP, into account in the numerical model performance evaluation. Based on value ranges of d

_{r}′, a classification into model performance ratings was proposed as well.

_{x}, p, and F

_{x}), which demonstrates OF’s applicability for the design of such hybrid coastal defence systems. Remaining discrepancies were found to be mainly caused by the different wave generation methods applied in OF (static boundary) and EXP (moving wave paddle), which caused an underestimation of the incident wave energy and an overestimation of the wave setup in OF compared to EXP. Consequently, when applying OF for a design of a hybrid coastal defence system, the incident wave energy is recommended to be calibrated, while the wave setup development for a static boundary condition with AWA in OF is actually closer to the field condition compared to EXP (finite water mass).

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Numerical Convergence Analysis

#### Appendix A1. Model Convergence Statistics

_{c}is the freeboard, and B is the bias defined by (7). The bias or difference in the wave setup is normalised with the freeboard which is one of the governing parameters for waves overtopping a dike [83].

_{max}and O

_{min}are the respective maximum and minimum values of the reference time series.

#### Appendix A2. Convergence Analyses

_{sez}up to the dike toe (i.e., β

_{sez}= 0, 1, 2, 3; Figure 3) and uses the mesh with the highest level (i.e., β

_{sez}= 3 or Δx = Δz = 0.0225 m) as the reference to which the other (coarser) resolution simulations are compared to. Convergence is achieved when no other significant changes are observed compared to a finer grid resolution model. The time stepping convergence analysis uses the run with the lowest maxCo number (i.e., maxCo = 0.15) as the reference to which other temporally coarser simulations (i.e., maxCo = 0.45, 0.25) are compared to. The statistical error indicators from Section A1 are provided in Figure A1 and Figure A2. All errors are close to or less than 5% at the toe of the dike for β

_{sez}= 2 (i.e., Δx = Δz = 0.045 m) and maxCo = 0.25. Even though maxCo = 0.45 does not show much higher errors than a value of 0.25, maxCo = 0.25 was preferred, because higher maxCo simulations were found to be prone to numerical instabilities. In any case, as long as the maxCo number cannot be defined separately for the air and water phases, the time step is mostly determined by the high spurious velocities that occur at the water-air interface. Because these spurious velocities are much higher (2–3 times) than the velocities in the water phase, much lower Courant numbers are actually obtained in the water phase [46]. This also explains why only limited differences between the tested maxCo values are observed here.

_{sez}= 2; maxCo = 0.25).

**Figure A1.**OF model grid resolution convergence analysis of the η time series at the WG locations along the flume up to the dike toe (WG14) based on (

**a**) the normalised bias, (

**b**) the residual normalised standard deviation, (

**c**) the residual correlation coefficient, and (

**d**) the normalised mean-absolute-error. The reference is the finest mesh with a refinement level in the surface elevation zones β

_{sez}of 3.

**Figure A2.**OF model time step convergence analysis based on maxCo for the mesh with β

_{sez}= 2. The reference is the lowest maximum Courant number applied (maxCo = 0.15). See caption of Figure A1 for the description of (

**a–d**).

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**Figure 1.**Overview of the geometrical parameters of the wave flume and WALOWA model set-up, with indicated wave gauge locations. Reprinted with permission from [29].

**Figure 2.**(

**a**) WGs deployed along the flume side wall to measure η; (

**b**) PWs; (

**c**) ECM to measure U

_{x}; (

**d**) WLDMs installed on the promenade to measure η; (

**e**) Hollow steel profile attached to two LCs and (

**f**) aluminium plate equipped with pressure sensors (PS) to measure F

_{x}and p.

**Figure 3.**Definition of the OF 2DV computational domain, with coloured indication of the model boundary types. The still water level (SWL) is indicated in blue (z = 4.14 m). The number in each of the mesh subdomains of the model domain (demarcated by black dotted lines) is the refinement level β applied in each subdomain (for β = 0, 1, 2, and 3: Δx = Δz = 0.18 m, 0.09 m, 0.045 m, and 0.0225 m). Note: the axes are in a distorted scale.

**Figure 4.**Comparison of the η time series at all sensor locations (

**a**–

**j**), including η

_{LW}in (

**a**–

**f**) (bold lines). The zero-reference is the SWL for (

**a**–

**f**) and the promenade bottom for (

**g**–

**j**).

**Figure 5.**Comparison of U

_{x}time series at all sensor locations (for the PWs in (

**a**,

**c**–

**e**); for the ECM in (

**b**)). The zero-reference is the promenade bottom at the sensor locations. For OF, both U

_{x}at the measured height above the promenade and the depth-averaged ${\overline{U}}_{x}$ time series are shown.

**Figure 6.**Comparison of F

_{x}time series for the vertical wall. The experiment is the LC force measurement.

**Figure 7.**Comparison of p time series for all vertical pressure sensor locations (for PS01–13 in (

**a**–

**m**)), PS01 being the bottom PS and PS13 the top-most PS.

**Figure 8.**Comparison of H

_{rms}between OF and (R)EXP up to the dike toe. From top to bottom: H

_{rms,sw}for the SW components, H

_{rms,lw}for the LW components, H

_{rms}for the total η, the wave setup $\overline{\eta},$ and finally an overview of the sensor locations, SWL, and bottom profile.

**Figure 9.**Index d

_{r}of REXP and OF with EXP up to the dike toe. From top to bottom: d

_{r,sw}for η

_{SW}, d

_{r,lw}for η

_{LW}, d

_{r,tot}for η, and finally an overview of the sensor locations, SWL, and bottom profile.

**Figure 10.**Comparison of R for η of REXP and OF with EXP up to the dike toe. From top to bottom: R

_{sw}for η

_{SW}, R

_{lw}for η

_{LW}, R for η, and finally an overview of the sensor locations, SWL, and bottom profile.

**Figure 11.**Index d

_{r}of REXP and OF with EXP from the dike toe up to the vertical wall. From top to bottom: d

_{r}for η and U

_{x}, and finally an overview of the sensor locations, SWL, and bottom profile.

**Figure 13.**Snapshots of selected key time instants chronologically over the first main impact (

**a**–

**f**). The OF snapshot (left) is compared to the equivalent EXP snapshot from the side view (centre) and top view (right) cameras. In the OF snapshots, the colours of the water flow indicate the velocity magnitude |U| according to the colour scale shown at the top. The red arrows are the velocity vectors, which are scaled for a clear visualisation. Each OF snapshot has two inset graphs: at the top is a time series plot of U

_{x}(for EXP and $\overline{{U}_{x}}$ for OF) (

**a**–

**c**) or F

_{x}(

**d**–

**f**), in which a circle marker (o) and a plus marker (+) indicate the time instant of the numerical and experimental snapshot, respectively. Along the vertical wall, U

_{x}(

**a**–

**c**) or p (

**d**–

**f**) is plotted at the respective ECM sensor location or each PS location (the vertical axis is z [m]). Along the promenade, four vertical grey dashed lines indicate the sensor locations on the promenade, of which the WLDM gauges are also visible in the experimental snapshots (topped by blue plastic bags). The location of the ECM is at the second vertical grey dashed line from the left. The time instant of the numerical snapshot is provided by t

_{OF}.

**Figure 14.**Snapshots of selected key time instants chronologically over the second main impact (

**a**–

**f**). See the caption of Figure 13 for further descriptions.

**Table 1.**Hydraulic parameters for the WALOWA bichromatic wave test (EXP) and its repetition (REXP): h

_{o}is the offshore water depth, h

_{t}the water depth at the dike toe, H

_{m0,o}the incident offshore significant wave height, R

_{c}the dike crest freeboard, f

_{i}the SW component frequency, a

_{i}the SW component amplitude, and β

_{m}(= a

_{2}/a

_{1}) the modulation factor.

TestID [-] | Duration [s] | h_{o} [m] | h_{t} [m] | h_{t}/H_{m0,o} [-] | R_{c} [m] | f_{1} [Hz] | a_{1} [m] | f_{2} [Hz] | a_{2} [m] | β_{m} [-] |
---|---|---|---|---|---|---|---|---|---|---|

Bi_02_6 (EXP) & Bi_02_6_R (REXP) | 209 | 4.14 | 0.43 | 0.33 | 0.117 | 0.19 | 0.45 | 0.155 | 0.428 | 0.951 |

**Table 2.**Proposed classification of the relative refined index of agreement d′

_{r}and corresponding rating.

d′_{r} Classification [-] | Rating |
---|---|

0.90–1.00 | Excellent |

0.80–0.90 | Very Good |

0.70–0.80 | Good |

0.50–0.70 | Reasonable/Fair |

0.30–0.50 | Poor |

(−1.00)–0.30 | Bad |

Location | REXP | OF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

B* [-] | σ* [-] | R [-] | d_{r} [-] | B* [-] | σ* [-] | R [-] | d_{r} [-] | d′_{r} [-] | Rating [-] | |

WG02 | −0.01 | 1.01 | 1.00 | 0.97 | 0.06 | 0.94 | 0.96 | 0.85 | 0.88 | Very Good |

WG03 | −0.01 | 0.99 | 1.00 | 0.97 | 0.05 | 0.92 | 0.95 | 0.85 | 0.87 | Very Good |

WG04 | −0.01 | 1.00 | 1.00 | 0.97 | 0.06 | 0.91 | 0.95 | 0.84 | 0.87 | Very Good |

WG07 | 0.01 | 1.00 | 1.00 | 0.97 | 0.06 | 0.94 | 0.94 | 0.84 | 0.87 | Very Good |

WG13 | 0.00 | 0.97 | 0.94 | 0.83 | 0.04 | 0.95 | 0.73 | 0.66 | 0.83 | Very Good |

WG14 | 0.00 | 1.00 | 0.98 | 0.92 | 0.05 | 0.89 | 0.91 | 0.82 | 0.90 | Very Good |

WLDM01 | −0.02 | 0.99 | 0.99 | 0.92 | −0.08 | 1.00 | 0.89 | 0.80 | 0.88 | Very Good |

WLDM02 | −0.02 | 1.01 | 0.99 | 0.92 | −0.05 | 1.01 | 0.91 | 0.82 | 0.89 | Very Good |

WLDM03 | 0.00 | 0.98 | 0.99 | 0.92 | −0.03 | 0.98 | 0.90 | 0.82 | 0.90 | Very Good |

WLDM04 | 0.01 | 0.97 | 0.98 | 0.92 | −0.00 | 1.00 | 0.87 | 0.79 | 0.87 | Very Good |

**Table 4.**Pattern and model performance statistics for all U

_{x}measurement locations on the promenade.

Location | REXP | OF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

B* [-] | σ* [-] | R [-] | d_{r} [-] | B* [-] | σ* [-] | R [-] | d_{r} [-] | d′_{r} [-] | Rating [-] | |

PW01 | 0.02 | 0.96 | 0.91 | 0.80 | −1.24 | 1.55 | 0.58 | −0.10 | 0.10 | Bad |

ECM | −0.02 | 1.05 | 0.87 | 0.81 | −0.25 | 0.94 | 0.73 | 0.63 | 0.82 | Very Good |

PW02 | −0.05 | 0.99 | 0.88 | 0.82 | −0.66 | 1.22 | 0.65 | 0.29 | 0.48 | Poor |

PW03 | −0.02 | 1.00 | 0.92 | 0.86 | −0.57 | 1.06 | 0.68 | 0.40 | 0.54 | Reasonable/Fair |

PW04 | −0.03 | 1.02 | 0.88 | 0.77 | −0.42 | 0.88 | 0.58 | 0.37 | 0.61 | Reasonable/Fair |

**Table 5.**Pattern and model performance statistics for all p (PS) and F

_{x}(LC) measurement locations.

Location | REXP | OF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

B* [-] | σ* [-] | R [-] | d_{r} [-] | B* [-] | σ* [-] | R [-] | d_{r} [-] | d′_{r} [-] | Rating [-] | |

PS01 | 0.00 | 1.00 | 0.98 | 0.92 | −0.14 | 0.84 | 0.84 | 0.80 | 0.88 | Very Good |

PS02 | −0.01 | 0.99 | 0.97 | 0.92 | −0.10 | 0.82 | 0.77 | 0.77 | 0.84 | Very Good |

PS03 | 0.00 | 1.00 | 0.96 | 0.91 | −0.13 | 0.75 | 0.71 | 0.75 | 0.83 | Very Good |

PS04 | 0.02 | 0.99 | 0.94 | 0.87 | −0.13 | 0.74 | 0.66 | 0.72 | 0.85 | Very Good |

PS05 | 0.01 | 1.00 | 0.96 | 0.91 | −0.11 | 0.75 | 0.61 | 0.69 | 0.78 | Good |

PS06 | −0.01 | 0.97 | 0.96 | 0.90 | −0.13 | 0.78 | 0.61 | 0.72 | 0.82 | Very Good |

PS07 | −0.01 | 0.93 | 0.95 | 0.89 | −0.17 | 0.76 | 0.53 | 0.67 | 0.78 | Good |

PS08 | −0.05 | 0.86 | 0.94 | 0.86 | −0.20 | 0.74 | 0.46 | 0.65 | 0.78 | Good |

PS09 | −0.07 | 0.88 | 0.93 | 0.85 | −0.25 | 0.78 | 0.39 | 0.61 | 0.76 | Good |

PS10 | −0.04 | 0.93 | 0.94 | 0.90 | −0.24 | 0.77 | 0.48 | 0.67 | 0.77 | Good |

PS11 | −0.04 | 0.91 | 0.94 | 0.88 | −0.33 | 0.57 | 0.37 | 0.63 | 0.75 | Good |

PS12 | −0.20 | 0.79 | 0.89 | 0.78 | −0.55 | 0.53 | −0.05 | 0.42 | 0.65 | Reasonable/Fair |

PS13 | −0.15 | 0.57 | 0.92 | 0.77 | −0.59 | 0.33 | 0.12 | 0.40 | 0.63 | Reasonable/Fair |

LC | 0.00 | 0.97 | 0.90 | 0.90 | −0.12 | 0.74 | 0.73 | 0.76 | 0.85 | Very Good |

Figure | Description | |
---|---|---|

Main Impact 1 | Figure 13a | Pre-impact of small overtopped wave. |

Figure 13b | Pre-collision of large overtopped bore and small wave reflected from vertical wall. | |

Figure 13c | Collision of large overtopped bore and reflected small wave. | |

Figure 13d | Impact on vertical wall of high velocity spray from overturned bore. | |

Figure 13e | Dynamic impact of overturned bore on vertical wall. | |

Figure 13f | Quasi-static impact of overturned bore on vertical wall. | |

Main Impact 2 | Figure 14a | Very small overtopped bore. |

Figure 14b | Impact of small overtopped bore on vertical wall. | |

Figure 14c | Impact of large overtopped bore on vertical wall. | |

Figure 14d | Impact of large overtopped bore on vertical wall, continued. | |

Figure 14e | Impact of large overtopped bore on vertical wall, continued. | |

Figure 14f | Return flow of large bore reflected from vertical wall. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gruwez, V.; Altomare, C.; Suzuki, T.; Streicher, M.; Cappietti, L.; Kortenhaus, A.; Troch, P.
Validation of RANS Modelling for Wave Interactions with Sea Dikes on Shallow Foreshores Using a Large-Scale Experimental Dataset. *J. Mar. Sci. Eng.* **2020**, *8*, 650.
https://doi.org/10.3390/jmse8090650

**AMA Style**

Gruwez V, Altomare C, Suzuki T, Streicher M, Cappietti L, Kortenhaus A, Troch P.
Validation of RANS Modelling for Wave Interactions with Sea Dikes on Shallow Foreshores Using a Large-Scale Experimental Dataset. *Journal of Marine Science and Engineering*. 2020; 8(9):650.
https://doi.org/10.3390/jmse8090650

**Chicago/Turabian Style**

Gruwez, Vincent, Corrado Altomare, Tomohiro Suzuki, Maximilian Streicher, Lorenzo Cappietti, Andreas Kortenhaus, and Peter Troch.
2020. "Validation of RANS Modelling for Wave Interactions with Sea Dikes on Shallow Foreshores Using a Large-Scale Experimental Dataset" *Journal of Marine Science and Engineering* 8, no. 9: 650.
https://doi.org/10.3390/jmse8090650