# Flow Depths and Velocities across a Smooth Dike Crest

^{*}

## Abstract

**:**

## 1. Introduction

_{r}, the instantaneous values of the overtopping flow depths h; and velocities u at different positions from the off-shore to the in-shore edge of the dike crest. The new data of h and u were used to derive a statistical representation of the overtopping flow characteristics over the structure crest.

## 2. Characterization of the Database

#### 2.1. Numerical Setup and Tested Conditions

_{s}(0.1 or 0.2 m) and various peak periods T

_{p}(ranging approximately from 1.3 to 6.5 s) specifically identified to test values of the wave steepness H

_{s}/L

_{m−}

_{1,0}between 0.02 and 0.05, where L

_{m−}

_{1,0}is the wave length computed from the spectral wave period T

_{m−}

_{1,0}.

_{off}) = 4 or 6) and fixed landward slope (cot(α

_{off}) = 3). The crest width (G

_{c}) and the structure height (h

_{c}) were kept constant and respectively equal to 0.3 m and 0.85 m. The seven crest freeboard values (R

_{c}), varying with the still water depth (wd), were selected to test positive, zero and negative freeboard conditions. The geometrical features and the wave characteristics were defined starting from the experiments on wave overtopping at dikes carried out in the small wave flume of the Leichtweiss Institute for Hydraulic Engineering of the Technical University of Braunschweig in Germany [4] and in the flume of WL|Delft Hydraulics in The Netherlands [5]. Those experiments represent, indeed, the basis of most of the literature studies aimed at characterizing the overtopping flow over the dike slopes and crest.

- Three wgs were placed at approximately 19 m from the wave generator to estimate the wave reflection coefficient K
_{r}based on the methodology by Zedlt and Skjelbreia [19]; - Thirty and 26 wgs were placed, respectively, on the off-shore slope (between 37.5 and 39 m from the wave generator, interspaced with a uniform interval of Δx = 0.05 m) and on the in-shore slope (from 39.3 to 40.3 m, with Δx = 0.04 m); these gauges were installed for analyzing the wave run-up and the wave overtopping and calculating the wave transmission coefficient;
- Eleven wgs (Δx = 0.03 m) were placed across the dike crest, from 39 to 39.3 m, to characterize the flow over the crest.

#### 2.2. Experimental Setup and Tested Conditions

_{s}= 0.06 m and L

_{m−}

_{1,0}= 3 m, while the water depth wd before the wave-maker may be at maximum of about 0.4 m.

_{s}in the range 0.04–0.06 m and wave periods T

_{p}≈ 0.85–1.51 s, realizing values of H

_{s}/L

_{m−}

_{1,0}≈ 0.03–0.04. The wave attacks were performed against four dikes, whose cross-sections differed for the crest widths, G

_{c}= 0.15 and 0.3 m, and/or the off-shore slope, cot(α

_{off}) = 2 and 4. Figure 2b shows two out of the four dike configurations, where cot(α

_{off}) = 2 is combined with G

_{c}= 0.15 m (left) and cot(α

_{off}) = 4 is combined with G

_{c}= 0.30 m (right). The remaining two configurations (not shown for sake of brevity) were realized by combining cot(α

_{off}) = 2 with G

_{c}= 0.30 m and cot(α

_{off}) = 4 with G

_{c}= 0.15 m. All the structures were 0.35 m high (see Figure 2b) and were positioned in the wave-flume so that the spot of the dike’s off-shore crest edge was always at the same distance of 10.75 m from the wave-maker, as indicated in Figure 2a. Each dike configuration was tested with three crest-freeboard conditions, R

_{c}/H

_{s}= 0, 0.5 and 1, obtained by varying the water depth wd in the channel from 0.29 to 0.35 m. For each freeboard, several wave attacks were performed, by varying the values of H

_{s}and T

_{p}, resulting in a total of 60 tests. All the experiments were performed in dry landward conditions.

- The 1st group includes three symbols, which may be “R00,” “R05” or “R10,” and refers to the target value of R
_{c}/H_{s}(R00 stands for R_{c}/H_{s}= 0; R05 stands for R_{c}/H_{s}= 0.5 and R10 stands for R_{c}/H_{s}= 1.0); - The 2nd group may be “H04,” “H05” or “H06” and each represents the target H
_{s}value (respectively 0.04, 0.05 and 0.06 m); - The 3rd group includes “s3” or “s4” and refers to the target wave steepness s
_{m-}_{1,0}= H_{s}/L_{m-}_{1,0}= 0.03 or 0.04; - The 4th group is “G15” or “G30,” which refer respectively to G
_{c}= 0.15 or 0.3 m; - The 5th group is “c2” or “c4” and refers to cot(α
_{off}) = 2 or 4, respectively.

- Three wgs, placed at approximately 1.5 times the maximum L
_{m−}_{1,0}from the wave-maker (≈5 m) to record the free-surface elevation with a sampling frequency of 100 Hz and separate the incident and reflected waves; the positions of the wgs are displayed in Figure 2a in red color. - Three ultrasonic Doppler velocity profilers (UVPs), which were installed along the structure crest and were used to record the time series of the vertical profiles of the horizontal flow velocities u and track the free surface elevation h. The positions of the three UVPs, shown in Figure 2b in green color and referenced as D1, D2 and D3, were selected to reconstruct the statistics of h and u in proximity of the dike crest off-shore edge (D1), in the middle of the crest (D2) and close to the in-shore edge (D3).
- A tank for the storage and the measurement of the overtopping volumes, placed at the end of the wave flume and below the channel and connected to recirculation system, regulated by a flowmeter (precision q = 1 × 10
^{−5}m^{3}/s), which collects the overtopped water from the tank and brings it back to the reservoir placed upstream the channel. - A 30 Hz full-HD camera to film the wave run-up and overtopping process; the camera was installed in front of the channel and corresponding to the upper part of the dike slope and the dike crest.
- All the structures were realized in a very smooth plexiglas material, which can be characterized by a roughness factor of γ
_{f}= 1 [13].

#### 2.3. Methodology for the Reconstruction of the Overtopping Flow Characteristics

- u. At each time step, the UVPs recorded a vertical profile of the radial velocity along the acoustic beam that was forming a 15°-angle (i.e., the Doppler angle) with the horizontal crest of the dike. The actual velocity was then reconstructed by assuming horizontal vectors (i.e., aligned flow with the dike crest). The range of the measured velocities depended on the settings of the probes; i.e., emitting frequency, pulse repetition period, the beam width and the Doppler angle. In the present experiments, the UVP settings yielded ~10 cm as the maximum layer thickness (as expected to occur above the dike crest during the largest wave attacks with H
_{s}= 0.06 m), with a profile resolution of 1.01 mm, and 4.2 m/s as the maximum velocity, with a nominal accuracy of 0.1% depending on the developing turbulence patterns in the test. - h. At each time step and with the same spatial resolution of the flow velocities, the UVPs also recorded the vertical profiles of echo (dB) of the acoustic impulse reflected by the particles transported by the water. In accordance with the free surface, the acoustic impulse undergoes a strong reflection induced by the density variation, which determines, in turns, a sharp peak of the echo value. The time series of the free surface elevations, and of the layer thicknesses h, were reconstructed based on the peak position in the instantaneous vertical profiles of the echo.

_{2%}and u

_{2%}). Based on the literature [5], these latter quantities were, respectively, the values of h and u exceeded by 2% of the incoming waves and were used by several authors [1,4,5,8] to characterize the extreme overtopping flow.

#### 2.4. Scale and Model Effects

^{3}–5 × 10

^{4}, which, rescaled to prototype conditions, correspond to Reynolds numbers in the range 5 × 10

^{5}–2 × 10

^{6}. Overall, reduced Reynolds numbers might determine higher drag coefficients, and consequently, smaller run-up heights and less overtopping at small scale [26]. However, the effects of increased drag forces on the overtopping due to scale effects are considered to be relevant for rubble mound structures, while they tend to be less effective or negligible for smooth structures [28,29].

## 3. Verification of the Data

_{r}in Section 3.1 and Section 3.2, respectively. For q, the EurOtop equations [28] and the artificial neural network (ANN) developed by the authors [31,32,33] were considered. The data of K

_{r}were instead compared to the formula by Zanuttigh and van der Meer [34] and to the ANN.

#### 3.1. Wave Overtopping Discharge

_{c}/H

_{s}≥ 0 and dry landward conditions is represented by the formulae of EurOtop (2018). For a probabilistic design and following the mean value approach, these formulae give:

_{m}

_{−1,0}is the Iribarren–Battjes breaker parameter and the factors γ

_{b}, γ

_{f}, γ

_{v}and γ

_{b}are the reducing coefficients for a berm, the roughness of the armor layer, the presence of a crown wall and oblique wave attacks. In the new numerical and experimental tests presented in this paper, all these factors are equal to 1. The two formulae Equations (1a) and (1b) are respectively valid for breaking (i.e., approximately for values of ξ

_{m}

_{−1,0}≤ 2) and non-breaking (ξ

_{m}

_{−1,0}> 2) wave conditions, where the wave breaking is supposed to occur for the interaction between the wave and the structure slope (α

_{off}).

_{c}/H

_{s}were computed from the incident wave heights H

_{s}, measured at the structural toe instead of the target values reported in Table 1 and Table 2. In both charts, only the tests at R

_{c}/H

_{s}≥ 0 and dry landward conditions are plotted. On average, the data are straightly and symmetrically distributed around the [28] curves. Most of the data fall within the 90% confidence bands associated to the predicting formulae (dashed lines in Figure 3). The greatest scatter is observed around the zero freeboard, where the formulae seem to underestimate part of the numerical and experimental data. Quantitatively, the agreement among the data and Equations (1a) and (1b) is provided for both the numerical and the experimental data in Table 3 in terms of the error indexes R

^{2}(coefficient of determination) and of σ

_{%}(relative standard deviation or coefficient of variation). Numerical data are significantly better predicted by Equations (1a) and (1b) than the experimental data, being σ

_{%}= 16% and R

^{2}= 0.95 in case of the numerical dataset and σ

_{%}= 56% and R

^{2}= 0.87 in case of the laboratory dataset. The lower agreement found for the experimental data is principally caused by the underestimation bias at R

_{c}/H

_{s}= 0, as it can be appreciated by Figure 3.

_{%}and R

^{2}are reported in Table 3. The agreement between the predictions and the measurements is remarkable in case of the laboratory dataset (σ

_{%}= 12% and R

^{2}= 0.91), as all the data are concentrated around the bisector line in Figure 4. The slight underestimation bias (μ(q

_{ANN}/q

_{lab}) = 0.89) could be explained with the particular smoothness of the dike material (plexiglas), which has been quantified with γ

_{f}= 1, whereas in the ANN training database the value of γ

_{f}= 1 is generally associated to concrete, asphalt, plywood, grass, etc. The values of q associated to the numerical dataset (σ

_{%}= 27% and R

^{2}= 0.81) are more scattered but still fairly represented by the ANN, as almost all the data fall within the 90% confidence bands (dashed lines in Figure 4) and no significant bias is observable.

#### 3.2. Wave Reflection Coefficient

_{r}resulting from the numerical simulations and the laboratory tests were compared to the predicting formula by Zanuttigh and van der Meer [34], which gave:

_{f}= 1. Equation (2) is valid for values of the wave steepness s

_{m−}

_{1,0h}≥ 0.01.

_{r}and the curve representing Equation (2) is provided in Figure 5a. In this chart, the quantity K

_{r}* is shown as a function of ξ

_{m-}

_{1,0}, where K

_{r}* = K

_{r}if R

_{c}/H

_{s}≥ 0.5, and K

_{r}* = K

_{r}/(0.67 + 0.37∙R

_{c}/H

_{s}) if −1 ≤ R

_{c}/H

_{s}< 0.5. All the data beyond the range of validity of Equation (2) have been removed from the plot.

_{m−}

_{1,0}> 2.5. The presence of a few “outliers,” i.e., data that fall outside the 90% confidence bands, can be explained by considering that Equation (2) represents an average trend and was fitted mostly on rubble mound data. Quantitatively, the agreement among the data and the formula is represented by σ

_{%}= 45% and R

^{2}= 0.77 in case of the laboratory tests and σ

_{%}= 28% and R

^{2}= 0.90 in case of the numerical tests, as reported in Table 3. These error indexes are in line with the uncertainty associated to Equation (2) (see [34]).

_{r}obtained with the ANN [33] is qualitatively given in Figure 5b and quantitatively characterized by the error indexes of Table 3. All the numerical and experimental tests fall within the range of applicability of the ANN. Both Figure 5b and Table 3 indicate that the data are slightly better represented by the ANN than Equation (2), with σ

_{%}= 40% and 6.0% and R

^{2}= 0.89 and 0.93 for the laboratory and numerical datasets, respectively. This can be explained with the higher number of parameters involved in the ANN with respect to the formula.

## 4. Validation of the Numerical Code

_{r}(Section 4.1) and of the flow depth h and velocities u at the dike crest (Section 4.2 and Section 4.3) were compared to the corresponding laboratory measurements.

#### 4.1. Wave Overtopping and Wave Reflection

_{m0}, T

_{m−}

_{1,0}and ξ

_{m−}

_{1,0}) measured in the laboratory (column “Lab”) and derived from the numerical model (column “Num”), including the dimensionless average values of q, q/(gH

_{m}

_{0}T

_{m−}

_{1,0}) and the values of K

_{r}. For each parameter, the mean μ and the standard deviation σ representative of the distribution of the ratio between numerical and experimental values (num/lab) are provided. Overall, the values of μ(num/lab) related to H

_{m}

_{0}, T

_{m−}

_{1,0}and ξ

_{m−}

_{1,0}, which are respectively equal to 0.87, 1.02 and 1.11, suggest that the code well reproduces the target wave conditions. The agreement between numerical and laboratory wave periods is remarkable (μ = 1.02, with σ = 8.4%), while the wave heights were slightly underestimated in the numerical code (μ = 0.87, σ = 8.0%). This result is line with the tendency of the IH-2VOF code to reduce the wave steepness by generating lower wave heights with respect to the target values, especially when the waves are generated in intermediate depth water (see, all of [15,16]), as in the present case. The lower values of s

_{m−}

_{1,0}obtained with the numerical code determined, in turn, slightly higher values of ξ

_{m−}

_{1,0}(μ = 1.11, σ = 9.75%); i.e., wave conditions slightly more distant from the wave breaking.

_{r}, the values of μ respectively equal to 0.94 and 0.96 indicate that on average the code can very well represent both the overtopping and the reflection processes, giving a very modest underestimation of both the quantities (μ < 1). The underestimations can be explained considering the different methodologies adopted in the numerical code and in the laboratory to calculate q and the different sampling frequencies used to record the free-surface elevations at the three wgs to reconstruct K

_{r}(100 Hz in the lab and 20 Hz in the numerical code). The lab values of q correspond to the average quantities measured from the tank (see Section 2.2), while the numerical values were derived from the integration of the values of u by the corresponding values of h recorded at the dike off-shore edge. Nevertheless, the differences between numerical and experimental values are on average 5%, with standard deviations of 16% (K

_{r}) and 28% (q), which are very limited and significantly lower than the average uncertainty associated to the common predicting methods (see, e.g., [34] or [35] for K

_{r}, and [28] for q).

#### 4.2. Water Depth Envelopes

#### 4.3. Extreme Flow Depths and Velocities

_{2%}and u

_{2%}calculated at the edges of the dike crest (precisely, at D1 and D3, see Figure 2b) are compared here to the corresponding values extracted from the elaboration of the echo and velocity profiles recorded from the UVPs installed at the same positions on the dikes in the laboratory. This analysis proposes a punctual verification of the numerical data, as it is not possible to draw the envelopes of the h and u-values from the UVPs measurements, which are available in accordance with the three UVPs only.

_{s}and (gH

_{s})

^{0.5}, respectively, to account for the different wave conditions generated in the numerical and laboratory flumes. The numerical values of h

_{2%}(Figure 8a) at both D1 and D3 are symmetrically distributed around the bisector line, representing the perfect association with the corresponding laboratory data. The agreement is represented by values of the standard deviation σ respectively equal to 0.006 m and 0.005 m for D1 and D3; the corresponding relative standard deviations (or coefficient sof variation) σ

_{%}are 26% and 30%.

_{2%}in Figure 8b are aligned directly with or above the bisector line, suggesting that, on average, the numerical code gives slightly overestimations of u

_{2%}. This tendency is observed at both D1 and D3 and can be quantified with a value of μ(u

_{2%,num}/u

_{2%,lab}) = 1.03 and 1.09 for D1 and D3, respectively. Overall, the standard deviation associated to the two distributions is σ = 0.09 m/s and 0.12 m/s at D1 and D3, respectively, with σ

_{%}= 9.8% at D1 and 13.7% at D3. The slight tendency to overestimate the u-values by the numerical code is probably due to the different representation of the friction with the dike surface and of the air entrainment. The IH2VOF code, indeed, is monophasic and cannot account for the air bubbles entrapped in the overtopping flow within the experiments (see Section 2.4). The air entrainment might induce a larger wave energy dissipation in the laboratory tests with respect to the numerical modelling and reduce, in turn, the experimental flow velocities.

## 5. Extreme Flow Depths and Velocities at the Dike’s Off-Shore Edge

_{2%}and u

_{2%}resulting from the numerical and experimental modelling are compared to the literature formulae. After discussing the performance of the existing methods and investigating the reasons of the discrepancies between data and formulae (Section 5.2), a new fitting is proposed to update the state-of-the-art formulae for the prediction of h

_{2%}and u

_{2%}at the off-shore edge.

#### 5.1. Comparison with Literature Formulae

_{2%}(x

_{c}= 0) = c

_{h}× (R

_{u}

_{,2%}− R

_{C}), R

_{C}≥ 0

_{2%}(x

_{c}= 0) = c

_{u}× [(R

_{u,}

_{2%}− R

_{C})]

^{0.5}, R

_{C}≥ 0

_{2%}and u

_{2%}are, respectively, the flow thicknesses and the flow velocities at the off-shore edge (x

_{c}= 0) of the dike crest that are exceeded by the 2% of the incident waves. The estimations of h

_{2%}and u

_{2%}by Equations (3) and (4) depend on R

_{u}

_{,2%}, which is the value of the wave run-up exceeded by the 2% of the waves and can be calculated based on EurOtop [28]. The values or the formulations proposed by the various authors for the coefficients c

_{h}and c

_{u}are summarized in Table 5. In the earliest formulae [4,5], the fitting coefficients c

_{h}and c

_{u}are constant values, while in more recent methods [1,8], c

_{h}and c

_{u}are functions of the off-shore slope α

_{off}. Moreover, while the formulations for u

_{2%}by Schüttrumpf [4] and Van Gent [5] were targeted to represent the flow velocities, the formulations by Bosman et al. [8] and by van der Meer et al. [1] aim at predicting the front velocities, or wave celerities, giving higher estimations of u

_{2%}.

_{2%}(x

_{c}= 0) and u

_{2%}(x

_{c}= 0) gathered with the new numerical and experimental modelling to the corresponding predicting formulae; i.e., Equations (3) and (4). In both Figure 9 and Figure 10, the results are grouped by values of cot(α

_{off}) and refer to tests in dry landward conditions R

_{c}/H

_{s}≥ 0. The data are further distinguished between experimental (filled-in symbols) and numerical (void symbols). The two charts of Figure 10 show the tests at R

_{c}/H

_{s}> 0 (panel a) and R

_{c}/H

_{s}= 0 (panel b). Following the structure of Equations (3) and (4), the results are shown as functions of (R

_{u}

_{,2%}− R

_{c}) in Figure 9 and of (g∙(R

_{u,}

_{2%}− R

_{c}))

^{0.5}in Figure 10.

_{2%}(Figure 10) are close to the curves by Bosman et al. [8]. It should be noted that Bosman et al. [8] provided fittings for cot(α

_{off}) = 4 and 6, exclusively (see Table 5). The curves for the prediction of h

_{2%}and u

_{2%}in case of cot(α

_{off}) = 2 have been extrapolated from the original formulae by Bosman et al. [8]. For brevity, the notations “c2,” “c4” and “c6” will be used in the following to refer to cot(α

_{off}) = 2, 4 and 6, respectively (see Section 2.2).

- All the values of h
_{2%}fall within the straight lines representing the theoretical formulae, with the exception of one test at c6 that slightly exceeds the upper line by Bosman et al. [8] for c6 (dot-dashed line). - In agreement with Bosman et al., all the data show a non-negligible effect of the structure slope: the milder the slope, the higher h
_{2%}. - The formulation by Van Gent [5]—which does not account for the slope effect—can be used to get an “average” estimation of the values of h
_{2%}. - For modest values of the wave run-up, i.e., for (R
_{u}_{,2%}− R_{c}) < 0.10–0.15, the formulae by Bosman et al. (2008), give an accurate representation of the data, while for (R_{u,}_{2%}− R_{c}) > 0.15, the formulae underestimate the values of h_{2%}in case of c2 and c4. The underestimation increases when increasing (R_{u}_{,2%}− R_{c}), and reaches ~100% when (R_{u,}_{2%}− R_{c}) ≈ 0.45 (data c4, circles in Figure 7). This might be in part explained by considering that the experimental tests used to calibrate the formulae were characterized by values of (R_{u,}_{2%}− R_{c}) ranging between 0 and 0.3. - Overall, the data seem to follow a non-linear trend with (R
_{u,}_{2%}− R_{c}) and a refitting of Equation (3) to extend its validity to the cases of c2 and (R_{u,}_{2%}− R_{c}) > 0.3 will be discussed in Section 4.2.

_{c}/H

_{s}and s

_{m-1,0}on h

_{2%}, without leading to any relevant result.

_{2%}(x

_{c}= 0), the analysis of Figure 10 leads to the following considerations.

- Similarly to h
_{2%}, the formula by Van Gent [5] (dotted line) gave an average estimation of the u_{2%}-values, representing, respectively, an upper and a lower envelope for the data at c4 and c6. The data at R_{c}= 0 (panel b)—which are out of the range of validity of the formula—were significantly over-estimated by Van Gent. - The effect of cot(α
_{off}) is still evident: the milder the slope, the higher u_{2%}(x_{c}= 0)—but slightly smoothed, with respect to h_{2%}(x_{c}= 0). With the exception of the data at c2, most of the data of u_{2%}were over-predicted by the formulae by [8], by 30%–50% in case of R_{c}/H_{s}> 0 (filled-in points) and of 40%–80% in case of R_{c}/H_{s}= 0 (void points). - On average, the data at R
_{c}/H_{s}= 0, tend to be lower than the data at R_{c}/H_{s}> 0 for the same value of the abscissa; i.e., (g∙(R_{u}_{,2%}− R_{c}))^{0.5}.

_{2%}values and the effects of cot(α

_{off}) and R

_{c}/H

_{s}, is given in the next Section 5.2.

#### 5.2. Discussion of the Results

_{2%}(x

_{c}= 0) can be explained considering the following three aspects.

_{u}based on the results of the previous experiments [4] and [5]. In those experiments, the measurements of u were obtained from micro-propellers placed at a fixed height over the structures. Therefore, the formulae were calibrated against punctual measures of u, while the values of u

_{2%}obtained from the new experimental and numerical modelling (and shown in Figure 10) were based on the average values of the velocities along the vertical above the structure crest. To further investigate this aspect, Figure 11 shows as example 2 instantaneous vertical profiles of u measured at D1 (i.e., approximately at x

_{c}= 0) during the test R00H05s3G15c4 conducted in the laboratory (Figure 11a) and reproduced with the numerical code (Figure 11b). Both the profiles refer to the instant of occurrence of the maximum flow depth h at D1 (i.e., around 0.38 m). Both the charts show a high variability of the u along the vertical: the vertically-averaged values of the u are 0.34 and 0.39 m/s for the laboratory and numerical test, respectively, with standard deviations of ±0.10 and ±0.05 m/s, and maximum values of u which are higher than the average of approximately 30% and 20%. This example indicates how much the vertical position of the instrument may affect the measured value of u, especially in case of laboratory tests, where the uncertainty related to the measurements is higher (as evident by comparing Figure 11a to Figure 11b).

_{u}—which were originally calibrated by Schüttrumpf [4] on the measurements of u—to account for the higher values of the c observed for the tests. Based on the observation that, for the same test, c tended to be greater than u, Bosman et al. increased the values of c

_{u}from 1.37 to 1.64 to achieve a more cautious approach. The formulations of c

_{u}as functions of cot(α

_{off}) further prompted by Bosman et al. were fitted again on the values of c instead of u.

_{2%}-values for the new experimental dataset. The wave celerities have been computed using the procedure for the identification and coupling of the overtopping waves developed by references [36,37]. The procedure was applied to the time series of the h-signals registered at D1 and D3, providing as outputs, the time lags for the waves to propagate from D1 and D3. The c-values were computed from the time lags, known the distance D1–D3. Figure 12 shows that the c-values are indeed higher than the corresponding u

_{2%}, and the discrepancy among c and u tends to increase with increasing u.

_{c}/H

_{s}. The on-average lower u

_{2%}-values observed at R

_{c}/H

_{s}= 0 (Figure 10b) than at R

_{c}/H

_{s}> 0 (Figure 10a) are due to the different characteristics of the flow in the two freeboard conditions. When R

_{c}/H

_{s}> 0, the dike crest is located above the mean water level in the wave run-up area and the waves go overtop the dike crest during the crest phase exclusively, resulting, thus, in positive—i.e., in-shore directed—values of u at x

_{c}= 0. On the contrary, when R

_{c}/H

_{s}= 0, the dike crest level is situated exactly in line with the mean water level; i.e., in the middle between the wave run-up and run-down area. In such conditions, the waves can reach and overtop the dike crest during the through phase also, determining a flow in regard to the dike edge which is intermittently in-shore and off-shore directed, with values of u that area alternatively >0 and <0. This phenomenon is clearly evident in the example of Figure 13, which reports the frequency distribution of all the u-values measured with UVP at D1 for the same experimental test (H05s3G30c4) conducted at R

_{c}/H

_{s}= 0 (R00) and R

_{c}/H

_{s}= 1 (R10). In this example, there was a predominance of negative u-values in case of R00, while for R10, the distribution clearly shifted towards positive u-values. The different overtopping flow conditions and the resulting different distributions of the u-values affected, in turn, the values of u

_{2%}.

#### 5.3. Refitting of the Formulae

_{2%}and u

_{2%}at the dike off-shore edge (x

_{c}= 0) are proposed:

_{2%}and (R

_{u,2%}− R

_{c}), and u

_{2%}and [g(R

_{u}

_{,2%}− R

_{c})]

^{0.5}through the introduction of the power coefficient b = 1.35. The coefficients c

_{h}and c

_{u}of Equations (3) and (4) are, respectively, replaced by the new coefficients a

_{h}and a

_{u}, whose formulations vary linearly with cot(α

_{off}), as indicated in Equations (5) and (6). The dimensions of a

_{h}and a

_{u}are, respectively, [m

^{0.65}] and [m

^{0.65}s

^{−0.65}].

_{c}≥ 0, Equation (6) is valid for R

_{c}> 0 only. As discussed in Section 5.2, the values of u

_{2%}follow different trends with [g(R

_{u}

_{,2%}− R

_{c})]

^{0.5}at positive and at zero freeboard. Since no further data were available from the literature for check, no fitting was proposed for u

_{2%}(x

_{c}= 0) at R

_{c}= 0.

_{2%}and u

_{2%}, respectively, and quantitatively represented by the error indexes R

^{2}and σ

_{%}reported in Table 6. This Table includes also the values of R

^{2}and σ

_{%}associated to the application of Equations (3) and (4) to the new data obtained by considering the formulations of c

_{h}and c

_{u}[8].

_{2%}(x

_{c}= 0) and u

_{2%}(x

_{c}= 0) for some tests of the datasets FlowDike1 and FlowDike2 [38,39]. These tests involved 2D and 3D wave attacks against smooth dikes characterized by different values of G

_{c}(0.6 and 0.7 m, model scale values) and of cot(α

_{off}) (3 and 6, respectively in FlowDike1 and FlowDike2). The following tests from FlowDike1 and FlowDike2 were not considered for the analyses:

- All the tests with missing records of either R
_{u2%}, h_{2%}(x_{c}= 0) or u_{2%}(x_{c}= 0); - The tests giving zero or negative overtopping discharge (considered unreliable);
- All the tests with wind velocity >10 m/s, as this fitting does not include the wind effect.

_{2%}(x

_{c}= 0) and u

_{2%}(x

_{c}= 0) from the remaining tests are compared to Equations (5) and (6) in Figure 14b and Figure 15b, respectively. The quantitative assessment of the agreement between the FlowDike1 and FlowDike2 data and the new formulae are given in Table 6, that also includes the performance of Equations (3) and (4) for comparison.

_{2%}(Figure 14) or of u

_{2%}(Figure 15). The best agreement is found between Equation (5) and the values of h

_{2%}from the numerical dataset, being that R

^{2}= 0.996 and σ

_{%}= 4.1% (see Table 6). A larger scatter was observed for the other datasets, as indicated by the values of σ

_{%}ranging between 15% (new laboratory data of u

_{2%}and Equation (6)) and 51% (FlowDike2 data of u

_{2%}and Equation (6)). The relatively high values of σ

_{%}are caused by the intrinsic scatter associated to the data, especially to the FlowDike1 and FlowDike2 experiments, and do not necessarily indicate a poor agreement with the new fitting. This is confirmed by the values of σ

_{%}associated to the predictions by Equations (3) and (4), which are comparable or even higher to the σ

_{%}associated to the new fitting. On the contrary, Figure 14 and Figure 15 reveal that the new formulae provide a good, “average” representation of most of the data, which are randomly but symmetrically distributed around the curves following their trends. This qualitative analysis is confirmed and reinforced by the values of R

^{2}of Table 6, which are always >0.72 and in six cases out of eight, even >0.80. Equations (3) and (4) provide a similar—but lower—performance on FlowDike1 and FlowDike2 data, while they give a worse representation of the new experimental and numerical data, as indicated by Figure 9 and Figure 10.

_{off}) = 2, 3) and low freeboards. Its use is suggested especially in case of relatively high run-up levels, i.e., approximately for values of (R

_{u,}

_{2%}−R

_{c})/H

_{s}> 2–2.5, which represent very severe surge conditions to catastrophic flooding scenarios.

_{c}/H

_{s}≤ 4.0 and 0 < R

_{c}/H

_{s}≤ 4.0 in case of Equation (5) and Equation (6), respectively; cot(α

_{off}) = 2; 3; 4; 6; 0.72 ≤ ξ

_{m-}

_{1,0}≤ 6.13; 0.035 ≤ H

_{s}≤ 0.22 m.

## 6. Evolution of Flow Characteristics along the Dike Crest

_{2%}and u

_{2%}and the literature formulae [7]. Section 6.2 and Section 6.3 are dedicated to an in-depth analysis of the values and trends of the flow velocities, with specific attention to the effects of the crest freeboard and wave breaking. Section 6.4 draws some conclusions about the analyses of the flow velocities, providing a few guidelines for practical use.

#### 6.1. Comparison with Literature Formulae

- The dike crest is horizontal;
- The vertical velocities can be neglected;
- The pressure term is almost constant over the dike crest;
- The viscous effects along the flow direction are small;
- The bottom friction is constant over the dike crest.

_{c}is the horizontal coordinate along the crest, h(x

_{c}) is the overtopping flow depth on the dike crest at the coordinate x

_{c}; u(x

_{c}) is the overtopping flow velocity on the dike crest at the coordinate x

_{c}; c

_{3}is a dimensionless coefficient, varying according to the quantile used for h and u (50%, 10%, 2%); and f is the bottom friction coefficient.

_{c}= 0.3 m and R

_{c}≥ 0. Therefore, for a smooth dike, the authors suggested the use of the (very low) value of f = 0.0058, which reduces u approximately 8%.

_{m-}

_{1,0}, according to the following relationship:

_{U}, derived from the continuity equation of Q, but no validation is given for this formula.

_{c}< 0. Moreover, the formulae for u are verified against the off-shore and the in-shore values u(x

_{c}= 0) and u(x

_{c}= G

_{c}) only. The numerical simulations (see Table 1) provide instead, continuous records of u and h between x

_{c}= 0 and x

_{c}= G

_{c}for any crest emergence and submergence. Due to the limits of the laboratory equipment, the following analyses and discussion are principally based on numerical results: experimental evidence is available for the values of u and h corresponding to the UVPs; i.e., at D1, D2 and D3, for R

_{c}/H

_{s}≥ 0 and dry landward conditions.

_{c}/H

_{s}. In case of R

_{c}/H

_{s}< 0, only tests in dry landward conditions were considered.

_{2%}(panel a) decays almost linearly with x

_{c}, showing a certain dependency with R

_{c}/H

_{s}. The decrease of h, ranging between 15 and 20% for R

_{c}< 0, and being 35% for R

_{c}/H

_{s}= 1.5, is in agreement with EurOtop [28] but it is lower than the estimations of Equation (7), which would predict a decrease of approximately the 50%. The same decays were approximately found when comparing the values of h

_{2%}(D1) with the corresponding values of h

_{2%}(D3) both in experiments and in simulations.

_{2%}by adopting the following values of the coefficient c

_{3}:

_{2%}(x

_{c}= G

_{c}) derived from the application of Equation (7) with the coefficients given in Equation (11) and the corresponding numerical values led to standard deviations σ and coefficients of determination R

^{2}, respectively equal to 0.06 and 0.93 for R

_{c}≥ 0 and to 0.09 and 0.96 for R

_{c}< 0. For R

_{c}< 0 in wet landward conditions, i.e., fully submerged or breached dikes, no significant changes of the values of h were detected along the crest.

_{2%}along the dike crest (Figure 16b) is significantly different for tests at positive or negative freeboard. In case of R

_{c}≥ 0, a slight decay was observable until x

_{c}/G

_{c}≈ 0.4, while the trend seems to invert around x

_{c}/G

_{c}= 0.6 and u increases up to approximately the same value of the crest beginning; i.e., u(x

_{c}= G

_{c}) ≈ u(x

_{c}= 0). In case of R

_{c}< 0, u monotonically increases with an apparent quadratic function of x

_{c}from the beginning to the end of the dike crest. The different trends of u were, therefore, investigated separately for positive and negative freeboards in Section 6.2 and Section 6.3, respectively.

#### 6.2. Flow Velocities at Zero and Positive Crest Freeboard

_{c}/G

_{c}> 0.4–0.5 was already observed by Guo et al. [17] in case of numerical tests with R

_{c}≥ 0. The combination of decreasing flow thicknesses (Figure 14a) and increasing flow velocities (Figure 14b) along the dike crest fulfills the continuity and the momentum balance equations, accounting also for the (small) effect of the friction. However, it is clearly in opposition with the existing approach by Schüttrumpf and Oumeraci [7] that is based on the approximation of the Navier–Stokes equations. Schüttrumpf and Oumeraci highlighted indeed, that their approach contradicts the instantaneous continuity equation, but they found experimental evidence of the decay of u and pointed out that the continuity equation is globally fulfilled by considering the time-integral of the product of u by h. Guo et al. [17] argued that the discrepancy between the results of their numerical modelling and the equations by Schüttrumpf and Oumeraci can be explained by a number of elements, such as (i) the dynamics of the overtopping flow over the dike crest and the shape of the water front; (ii) the effect of the air entrainment; and (iii) the limitations imposed by the assumptions by Schüttrumpf and Oumeraci; specifically, the approximation of the boundary layer and the adoption of flow-depth integrated velocities. These aspects are analyzed in detail in the following Section 6.2.1 and Section 6.2.2.

#### 6.2.1. Effect of the Wave Breaking

_{2%}hardly decays while propagating along the dike crest. Similarly to the results of Guo et al., the numerical model suggests that the flow velocity tends on the contrary to accelerate. The shape of the overtopping tongue is correctly represented by the numerical code (Figure 17b). However, the large amount of air pockets characterizing the flow in the laboratory (Figure 17a) cannot be captured by the mono-phase numerical code.

- From the off-shore edge to the impinging jet section, x
_{c}≈ [0; 0.4∙G_{c}], the overtopping tongue dissipates its energy in the change of direction from the up-rush along the seaward slope to the horizontal stream over the crest; - Corresponding to the section of the impinging jet (x
_{c}≈ 0.4∙G_{c}), the wave front hits violently against the dike crest surface and breaks; this section is subjected to the maximum impact (wave pressure), and as a consequence of the momentum balance equation, to the minimum velocity; the section is, therefore, associated to the maximum stress and possibly to the maximum scour risk; - In the second half of the dike crest, x
_{c}≈ [0.5∙G_{c}; G_{c}], the overtopping flow velocity tends to accelerate into a supercritical stream for the free-outfall boundary condition at the landward edge, while the potential energy accumulated at the hit turns into kinetic.

_{c}≈ [0.35; 0.4∙G

_{c}]. Based on the color map of Figure 16a, u is maximal at the off-shore edge and decelerates while the wave propagates along the dike crest. At the impinging section, the u-value is minimal, correspondingly with the crest surface, and it starts increasing again after the impact. The result of this process is the decreasing/increasing trend of u

_{2%}leading to values of u

_{2%}(x

_{c}= G

_{c}) ≈ u

_{2%}(x

_{c}= 0), as shown in Figure 16b.

_{2%}at x

_{c}= G

_{c}, which in the practice represents one of the most relevant parameters, as it governs the down-wash streaming in the inland area. Figure 19 compares the values of u

_{2%}(x

_{c}= G

_{c}) to the corresponding values of u

_{2%}(x

_{c}= 0) obtained from the experimental (Figure 19a) and numerical tests (Figure 19b) for the different breaking or non-breaking conditions. All the numerical tests (at R

_{c}/H

_{s}≥ 0) represent waves reaching the dike crest in breaking (cotα

_{off}= 4) or broken (cotα

_{off}= 6) conditions, while the experimental tests present both non-breaking (cotα

_{off}= 2) and breaking (cotα

_{off}= 4) wave conditions.

_{2%}effectively decays from x

_{c}= 0 to x

_{c}= G

_{c}only in case of c2; i.e., of non-breaking waves (green diamonds). The average decay was of approximately the 30%, ranging between 13% and 45%. The variability of the decay rate was due to the different combinations of crest widths, freeboard conditions, wave steepness and wave heights. However, no explicit or systematic relationship was found between the decay rate and any of the parameters G

_{c}, R

_{c}/H

_{s}, H

_{s}/L

_{m−}

_{1,0}and H

_{s}. The absence of a direct link between the decay rate and G

_{c}can be explained with the small friction determined by the smooth dike surface and by the limited value of G

_{c}. This result is in agreement with the synthesis by EurOtop [13,28], where a specific indication of negligible decay for flow depths larger than 0.1 m and G

_{c}around 2–3 m is reported.

_{2%}along the dike crest is almost negligible, and in some cases u

_{2%}(x

_{c}= G

_{c}) might result even greater than u

_{2%}(x

_{c}= 0), as already observed by Guo et al. [17]. Generally, the phenomenon of the increasing velocities from x

_{c}= 0 to x

_{c}= G

_{c}is more frequently observed with the numerical code (Figure 19b), which also tends to give a slightly lower decay rates with respect to the physical experiments (Figure 19a).

#### 6.2.2. Effect of the Crest Emergence

_{c}≥ 0, the decay rate of u

_{2%}from x

_{c}= 0 to x

_{c}= G

_{c.}was not directly correlated to R

_{c}/H

_{s}(see Section 6.2.1 and Figure 16b). Nevertheless, it was observed that the statistical distribution of the flow velocities was significantly different in cases of R

_{c}/H

_{s}= 0 and R

_{c}/H

_{s}> 0. To illustrate those outcomes, Figure 20 shows the frequency histograms of the instantaneous depth-averaged u-values measured during an example test case from the laboratory experiment H05s3G30c4 conducted at R

_{c}/H

_{s}= 0 (Figure 20a) and R

_{c}/H

_{s}= 1.0 (Figure 20b). In both panels of Figure 20a,b, the histograms are provided for the values of u recorded at both D1 (blue shading) and D3 (orange shading); viz., at x

_{c}= 0 and x

_{c}= G

_{c}, respectively. The probability density function (pdf) for R

_{c}/H

_{s}= 0 (Figure 20a) presents a sharp peak corresponding of the lowest values of u (the mode is around ≈0.03 m/s for both x

_{c}= 0 and x

_{c}= G

_{c}); then it decreases almost monotonically with the increasing u-values. The distribution for R

_{c}/H

_{s}= 1.0 (Figure 20b) was more flat and symmetrical, showing increasing and decreasing data frequencies for values of u, respectively, lower and greater than the mode, which is ~0.10 m/s for x

_{c}= 0 (blue bars) and ~0.10 m/s for x

_{c}= G

_{c}(orange bars). The different mode values in case of R

_{c}/H

_{s}= 1.0 indicate that higher values of u are more frequently detected at D3 than at D1 (i.e., the distributions of u at D3 are shifted towards higher values than the distributions at D1); i.e., at x

_{c}= G

_{c}rather than at x

_{c}= 0. In other terms, the flow more frequently accelerates than decelerates along the dike crest. This phenomenon is not detectable at R

_{c}/H

_{s}= 0 (Figure 20a).

- On average, the flow velocities at both the dike off-shore and in-shore edges are higher at R
_{c}/H_{s}> 0 than at R_{c}/H_{s}= 0, as already observed for the extreme percentiles u_{2%}reported in Figure 10; - In case of R
_{c}/H_{s}> 0, the u-values are more narrowly distributed around the mode, showing a lower variability with respect to the case at R_{c}/H_{s}= 0; - At R
_{c}/H_{s}> 0, the flow more frequently accelerates than decelerates from x_{c}= 0 to x_{c}= G_{c}, in line with the discussion proposed in Section 6.2.1.

#### 6.3. Flow Velocities at Negative Freeboard

_{2%}from x

_{c}= 0 to x

_{c}≈ 0.4G

_{c}observed for R

_{c}/H

_{s}≥ 0 no longer occurred at R

_{c}< 0 (see Figure 16b), because the energy dissipation induced by the wave breaking and jumping against the dike crest (see Figure 18 and Section 6.2.1) was reduced or nullified by the presence of the water over the crest. Indeed, at R

_{c}< 0, a weir-like constant supercritical flow was established over the dike crest due to the hydraulic gradient between the dike off-shore and in-shore edges. The free outfall condition at the landward slope governed the overflow process, inducing the flow to accelerate and resulting into values of u(x

_{c}= G

_{c}) greater than the values u(x

_{c}= 0). Figure 16b shows, indeed, a monotonic increasing trend of u

_{2%}from x

_{c}= 0 to x

_{c}≈ G

_{c}for all the cases at R

_{c}< 0.

_{c}/H

_{s}≥ 0 no longer applies. On the contrary, a sharp effect of R

_{c}/H

_{s}was observed, as the increased rate of u

_{2%}increased with the crest submergence. Such an effect was already evident from the average trends of Figure 16b and is displayed by Figure 21, which compares the numerical values of u

_{2%}(x

_{c}= 0) with the corresponding values u

_{2%}(x

_{c}= G

_{c}) for all the available numerical tests at R

_{c}< 0 in wet (panel a) and dry (panel b) landward conditions. In that figure, the data are clustered on different levels above the bisector line according to the values of R

_{c}/H

_{s}: the lower the value of R

_{c}/H

_{s}, the farther the data from the bisector; i.e., the lower the R

_{c}/H

_{s}, the larger the ratio u

_{2%}(x

_{c}= G

_{c})/u

_{2%}(x

_{c}= 0). On average, the values of u

_{2%}(x

_{c}= G

_{c})/u

_{2%}(x

_{c}= 0) ranged from 5% to 30% for R

_{c}/H

_{s}ranging from −0.2 to −1.5 in dry landward conditions. Similar results were observed for the wet landward conditions, though the increase of u

_{2%}from x

_{c}= 0 to x

_{c}= G

_{c}seemed to be more modest (up to 20%–25%) and not sharply related to R

_{c}/H

_{s}. This was probably due to the presence of the water on the landward side, which determines lower hydraulic gradients between the off-shore and in-shore edges, and consequently, more modest flow acceleration.

_{c}> 0.

#### 6.4. Remarks on the Flow Velocities and Design Recommendations

_{c}> 0; for larger crests and/or negative freeboards, the application of the existing formulae (Equations (8) or (9)) might lead to incautious estimates.

_{c}= 0) = u(x

_{c}= G

_{c}).

_{c}≥ 0 and non-breaking waves, the minimum or zero velocity is found roughly around the middle section of the dike crest, where the imping jet hits the dike surface, and therefore, the highest loads and stresses are concentrated in this area. Based on this finding, the section subjected to the impinging jet can be considered the most exposed to the risk of scour. That outcome may result in practical interest in the case of permeable structures, and can be combined to the analyses of van der Meer et al. [40], who correlated the start of the scour to the occurrence of a given critical flow velocity.

_{c}< 0, the maximum flow velocities are found at the dike in-shore edge. This may result in higher downwash velocities along the dike landward slope and in the landward area. Therefore, it is advised to increase, by approximately 20%, the value of u at the in-shore edge with respect to the value at the off-shore edge (see Figure 16b and Figure 21).

_{c}< 0 is available from numerical modelling only and that for R

_{c}≥ 0, the experimental data relate to punctual measurements of h and u in regard to the dike crest edges and its middle section.

## 7. Conclusions

_{off}) = 2, 4 and 6), relative crest freeboards in the range R

_{c}/H

_{s}= [−1.5; 1.5] and two crest widths G

_{c}(3 and 6 m, in prototype units). The combinations of H

_{s}/L

_{m}

_{-1,0}and cot(α

_{off}) values were such to determine both breaking (ξ

_{m−}

_{1,0}≤ 2) and non-breaking (ξ

_{m}

_{−1,0}> 2) wave conditions within both the numerical and the experimental tests.

_{r}) resulting from the new laboratory and numerical tests, with the corresponding predictions obtained from consolidated methods available from the literature ([28] for q; [34] for K

_{r}; the artificial neural network by [32,33] for both q and K

_{r}). The numerical model was validated by reproducing a subset of 12 experimental tests and by comparing the results in terms of: q, K

_{r}, maximum and mean water depths (h) envelopes and extreme flow depths and velocities (h

_{2%}and u

_{2%}) across the dike crest.

_{2%}and u

_{2%}at the dike off-shore edge (x

_{c}= 0) and the comparison with the existing predicting methods [4,5,8] led to the prompting of two new formulations, Equations (5) and (6). These formulae were conceived i) to achieve a better accuracy in the representation of the data; ii) correct some incautious underestimations or excessive overestimations associated to the existing formulae; and iii) extend the field validity of the existing formulae to structures with cot(α

_{off}) = 2 and R

_{c}= 0. Equations (5) and (6) follow the same formulation of the existing methods (Equations (3) and (4)) but represent a non-linear increase of h

_{2%}(x

_{c}= 0) and u

_{2%}(x

_{c}= 0) with (R

_{u}

_{,2%}− R

_{c}) and (g∙(R

_{u}

_{,2%}− R

_{c}))

^{0.5}, respectively. Equations (5) and (6) propose also two new coefficients a

_{h}and a

_{u}, which are proportional to cot(α

_{off}). The ranges of validity of the method are: cot(α

_{off}) = 2–6, ξ

_{m}

_{-1,0}= 1–4 and R

_{c}≥ 0 for h

_{2%}(x

_{c}= 0) (Equation (5)); and R

_{c}> 0 for u

_{2%}(x

_{c}= 0) and smooth structures, i.e., γ

_{f}= 1 (Equation (6)).

_{u,}

_{2%}− R

_{c})/H

_{s}< 2). Higher run-up heights were also modelled with the numerical code to analyze more severe or catastrophic scenarios ((R

_{u,}

_{2%}− R

_{c})/H

_{s}> 2–2.5) in a climate change situation. The validity of Equations (5) and (6) were checked against the two sets of experiments, FlowDike1 and FlowDike2 [39,40], on dikes with cot(α

_{off}) = 3 and 6. The new formulae are characterized by at least the same accuracy of the existing methods when applied to FlowDike1 and FlowDike2, while they provide a remarkably better representation of the new experimental and numerical data.

_{2%}and u

_{2%}along the dike crest, i.e., from the off-shore (x

_{c}= 0) to the in-shore (x

_{c}= G

_{c}) edge, were analyzed in detail from the numerical model at any crest emergence (−1.5 ≤ R

_{c}/H

_{s}≤ 1.5) and for both dry, landward conditions. The numerical results are supported by experimental evidence in correspondence of x

_{c}= 0, x

_{c}≈ 0.5 and x

_{c}= G

_{c}and for values of R

_{c}/H

_{s}≥ 0 and dry landward conditions. The analysis resulted in the following criteria for design application.

_{c}= 0 to x

_{c}= G

_{c}. The decay of h is strictly dependent on the crest freeboard, varying approximately between the 20% and the 35% for over-washed (R

_{c}/H

_{s}= −1.5) and for emerged (R

_{c}/H

_{s}= 1.5) conditions respectively. To account for the effect of R

_{c}/H

_{s}, a new coefficient c

_{3}for the decay formulation of h was proposed in Equation (10). For submerged dikes (R

_{c}< 0 and wet landward conditions), the decay of h is almost negligible.

_{c}and by the wave breaking or non-breaking conditions.

_{c}/H

_{s}≥ 0 and non-breaking wave conditions, u decreases in the first part, reaches a minimum around the half of the crest, in accordance with the impinging jet hitting the dike surface, and then, increases in the second part, being that the values of u at the landward edge are approximately equal to the seaward edge. The section subjected to the impinging jet is the most exposed to the risk of scour. For a conservative approach, in case of R

_{c}/H

_{s}≥ 0 it is suggested to assume u(x

_{c}= 0) = u(x

_{c}= G

_{c}), while the use of the decay trends for u (Equation (8)) is discouraged.

_{c}/H

_{s}≥ 0 and breaking or broken waves, u hardly changes along the dike crest. The overtopping flow is governed by the free-outfall boundary condition at the dike off-shore edge, which determines a supercritical accelerated weir-like flow, resulting in values of u(x

_{c}= G

_{c}) which may be even higher than u(x

_{c}= 0).

_{c}/H

_{s}< 0 u increases along the crest, and its growth rate increases by increasing the submergence, up to approximately the 30% for R

_{c}/H

_{s}< −1, both in the case of wet and dry landward conditions. Overall, it is suggested to assume a value of u at x

_{c}= G

_{c}which is increased of the 20% with respect to the value at x

_{c}= 0.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Layout of the numerical flume, including the dike’s cross-section and the position of the wage gauges (wgs), and references to the adopted symbols. Measures in m.

**Figure 2.**Panel (

**a**): scheme of the wave flume and layout of the equipment. Panel (

**b**): cross-sections of the dikes (slopes cot(α

_{off}) = 2 and 4, G

_{c}= 0.15 and 0.3 m) with reference to the position of the ultrasonic velocity profilers (UVPs) along the dike crest (D1, D2 and D3). Measures in m.

**Figure 3.**Experimental and numerical dimensionless values of q as functions of R

_{c}/H

_{s}compared to the curves representing the formulae by EurOtop (2018) for the prediction of q, Equations (1a) and (1b), in panels (

**a**) and (

**b**), respectively. The dashed lines represent the 90% confidence bands associated to the formulae. Data at R

_{c}/H

_{s}≥ 0 and dry landward conditions.

**Figure 4.**Comparison among experimental and numerical values of q and corresponding predictions from the artificial neural network (ANN). The dashed lines represent the 90% confidence bands associated to the formulae. Data in dry landward conditions.

**Figure 5.**(

**a**) Experimental and numerical values of K

_{r}as functions of ξ

_{m}

_{−1,0}compared to the curve representing Equation (2) for the prediction of K

_{r}. (

**b**) Comparison among experimental and numerical values of K

_{r}and corresponding predictions from the ANN. The dashed lines represent the 90% confidence bands associated to the formula and the ANN predictions.

**Figure 6.**Layouts of the numerical channel set up for the validation of the code. Structure configuration with G

_{c}= 0.30 and 0.15 m to the top and to the bottom, respectively.

**Figure 7.**Maximum and mean depth (h) envelopes along the dike crest (x

_{c}/G

_{c}) as obtained from the image processing (dotted-dashed lines, “video”) and the numerical code (dots, “num”). (

**Left**) Test R00H05s3G30c4; (

**right**) test R10H05s3G30c2.

**Figure 8.**Laboratory (abscissa) and numerical (ordinate) values of h

_{2%}(panel (

**a**)) and u

_{2%}(panel (

**b**)) calculated at D1 (off-shore edge, diamonds) and D3 (in-shore edge, squares).

**Figure 9.**Comparison among the values of h

_{2%}recorded at x

_{c}= 0 within the new experimental and numerical tests and literature formulae. The data refer to dry landward conditions and R

_{c}/H

_{s}≥ 0, are grouped by values of cot(α

_{off}) and are distinguished between experimental (filled in) and numerical (void).

**Figure 10.**Comparison among the values of u

_{2%}recorded at x

_{c}= 0 within the new experimental and numerical tests and literature formulae. The data refer to dry landward conditions and are grouped by values of cot(α

_{off}) and distinguished between experimental (filled in) and numerical (void). (

**a**) Tests at R

_{c}/H

_{s}> 0; (

**b**) tests at R

_{c}/H

_{s}= 0.

**Figure 11.**Instantaneous vertical profiles (z) of the flow velocity (u) measured at D1 (≈x

_{c}= 0) for the test R00H05s3G15c4 in the laboratory (panel (

**a**)) and reproduced with the numerical code (panel (

**b**)). The profiles refer to the instant of occurrence of the maximum flow depth at D1.

**Figure 12.**Comparison among values of u

_{2%}at the dike off-shore edge (x

_{c}= 0, D1) and average wave celerities c. The data belong to the experimental dataset and are grouped by values of cot(α

_{off}).

**Figure 13.**Frequency distribution of the values of u measured at D1 (≈x

_{c}= 0) for the same experimental test H05s3G30c4, conducted at zero (R00) and in emerged freeboard (R10) conditions. The flow is in-shore and off-shore directed when the values of u are, respectively, >0 and <0.

**Figure 14.**Comparison among the values of h

_{2%}recorded at x

_{c}= 0 during the new tests (

**a**) and the FlowDike data experiments (

**b**) and the new fitting by Equation (5). The data are grouped by values of cot(α

_{off}).

**Figure 15.**Comparison among the values of u

_{2%}recorded at x

_{c}= 0 during the new tests (

**a**) and the FlowDike data experiments (

**b**) and the new fitting by Equation (6). The data are grouped by values of cot(α

_{off}) and relate to R

_{c}> 0 only.

**Figure 16.**Evolution of the flow thickness (h

_{2%}/h

_{2%,xc}= 0, panel (

**a**)) and of the flow velocity (u

_{2%}/u

_{2%,xc}= 0, panel (

**b**)) along the dike crest (x

_{c}/G

_{c}). Average values from the numerical tests grouped by R

_{c}/H

_{s}.

**Figure 17.**Frames of an overtopping event propagating over the dike crest during the same test R00H05s3G30c4, relative to a breaking wave, and carried out in the laboratory (

**a**) and by the numerical code (

**b**). The color map displays the computational field of the u-velocity (m/s).

**Figure 18.**Consecutive frames of an overtopping event propagating over the dike crest during a numerical (

**a**) and an experimental (

**b**) test (R00H05s2G30c2) relative to non-breaking wave conditions. The color map of panel (

**a**) displays the u-velocity field (m/s).

**Figure 19.**Comparison among values of u

_{2%}measured at the off-shore (x

_{c}= 0) and in-shore (x

_{c}= G

_{c}) edges of the dike crest in the lab experiments (

**a**) and the numerical simulations (

**b**). The data are grouped by values of cot(α

_{off}). All the tests are at R

_{c}/H

_{s}≥ 0.

**Figure 20.**Frequency distribution of the instantaneous depth-averaged values of u at the dike off-shore (blu) and in-shore (orange) edges for the same experimental test (H05s3G30c4) at R

_{c}/H

_{s}= 0 (panel (

**a**)) and R

_{c}/H

_{s}> 0 (panel (

**b**)).

**Figure 21.**Comparison among values of u

_{2%}(x

_{c}= 0) to corresponding values of u

_{2%}(x

_{c}= G

_{c}) for wet (

**a**) and dry (

**b**) landward conditions. The data belong to the numerical database and are grouped by values of R

_{c}/H

_{s}. For all the data, R

_{c}/H

_{s}< 0.

**Table 1.**Synthesis of the range of configurations tested in the numerical model. The “dry” landward conditions were carried out with H

_{s}= 0.2 m and H

_{s}/L

_{m−}

_{1,0,t}= 0.02 and 0.03 only.

R_{c}/H_{s} | −1.5 | −1 | −0.5 | −0.2 | 0 | +0.5 | +1 | +1.5 |
---|---|---|---|---|---|---|---|---|

H_{s}/L_{m}_{−1,0,t} | 0.02; 0.03 | 0.02; 0.03; 0.04 | 0.02; 0.03; 0.05 | 0.02 | 0.02; 0.03; 0.04; 0.05 | 0.02; 0.03; 0.04 | 0.02; 0.03; 0.05 | 0.03 |

H_{s} (m) | 0.1; 0.2 | 0.1; 0.2 | 0.1; 0.2 | 0.2 | 0.1; 0.2 | 0.1; 0.2 | 0.2 | 0.2 |

cot(α_{off}) | 4; 6 | 4; 6 | 4; 6 | 4; 6 | 4; 6 | 4; 6 | 4; 6 | 4; 6 |

cot(α_{in}) | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |

Landward | wet; dry | wet; dry | wet; dry | dry | wet; dry | wet; dry | dry | dry |

**Table 2.**Summary of the target conditions of the 60 experiments of wave overtopping at dikes performed in the Laboratory of Bologna.

R_{c}/H_{s} | 0 | +0.5 | +1 |
---|---|---|---|

s_{m−1,0} (%) | 3; 4 | 3; 4 | 3; 4 |

H_{s} (m) | 0.04; 0.05; 0.06 | 0.04; 0.05; 0.06 | 0.05; 0.06 |

wd (m) | 0.35 | [0.32; 0.325] | [0.29; 0.30] |

cot(α_{off}) | 2; 4 | 2; 4 | 2; 4 |

G_{c} (m) | 0.15; 0.30 | 0.15; 0.30 | 0.15; 0.30 |

Landward | dry | dry | dry |

Tot. # | 24 | 18 | 18 |

**Table 3.**Error indexes R

^{2}and σ

_{%}characterizing the agreement between the new laboratory/numerical data of q and K

_{r}and corresponding predictions from literature methods.

q | K_{r} | |||||||
---|---|---|---|---|---|---|---|---|

Dataset | Equations (1a)–(1b) | ANN | Equation (2) | ANN | ||||

σ_{%} | R^{2} | σ_{%} | R^{2} | σ_{%} | R^{2} | σ_{%} | R^{2} | |

Laboratory | 56% | 0.87 | 12% | 0.91 | 45% | 0.77 | 40% | 0.89 |

Numerical | 16% | 0.95 | 27% | 0.81 | 28% | 0.90 | 6.0% | 0.93 |

**Table 4.**Summary of the 12 experimental tests carried out at the Laboratory of Bologna and reproduced in the numerical code.

Test ID | H_{m}_{0}(m) | T_{m−}_{1,0}(s) | R_{c}/H_{s}(-) | cotα (-) | G_{c}(m) | ξ_{m−}_{1,0}(-) | q/(gH_{s}T_{m−}_{1,0})(-) | K_{r}(-) | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lab | Num | Lab | Num | - | - | - | Lab | Num | Lab | Num | Lab | Num | |

R00H05s3G15c4 | 0.047 | 0.042 | 1.048 | 1.083 | 0.00 | 4 | 0.15 | 1.52 | 1.68 | 7.80 × 10^{−3} | 7.6 × 10^{−3} | 0.16 | 0.20 |

R00H05s3G15c2 | 0.049 | 0.049 | 1.036 | 1.040 | 0.00 | 2 | 0.15 | 2.93 | 2.93 | 7.09 × 10^{−3} | 1.8 × 10^{−3} | 0.37 | 0.41 |

R00H05s3G30c4 | 0.046 | 0.042 | 1.048 | 1.190 | 0.00 | 4 | 0.30 | 1.52 | 1.81 | 7.02 × 10^{−3} | 7.5 × 10^{−3} | 0.17 | 0.18 |

R00H05s3G30c2 | 0.048 | 0.048 | 1.036 | 1.190 | 0.00 | 2 | 0.3 | 2.94 | 3.40 | 7.48 × 10^{−3} | 8.0 × 10^{−3} | 0.32 | 0.36 |

R05H05s3G15c4 | 0.053 | 0.044 | 1.076 | 1.105 | 0.50 | 4 | 0.15 | 1.46 | 1.64 | 2.54 × 10^{−3} | 2.5 × 10^{−3} | 0.34 | 0.26 |

R05H05s3G15c2 | 0.053 | 0.046 | 1.374 | 1.105 | 0.50 | 2 | 0.15 | 3.72 | 3.22 | 2.49 × 10^{−3} | 2.7 × 10^{−3} | 0.53 | 0.48 |

R05H05s3G30c4 | 0.048 | 0.044 | 1.076 | 1.105 | 0.50 | 4 | 0.30 | 1.53 | 1.64 | 2.49 × 10^{−3} | 2.4 × 10^{−3} | 0.28 | 0.26 |

R05H05s3G30c2 | 0.054 | 0.048 | 1.052 | 1.105 | 0.50 | 2 | 0.3 | 2.55 | 3.15 | 3.22 × 10^{−3} | 2.4 × 10^{−3} | 0.53 | 0.51 |

R10H05s3G15c4 | 0.056 | 0.042 | 1. | 1.040 | 1.00 | 4 | 0.15 | 1.37 | 1.59 | 8.02 × 10^{−4} | 9.3× 10^{−4} | 0.41 | 0.27 |

R10H05s3G15c2 | 0.057 | 0.045 | 1.008 | 1.040 | 1.00 | 2 | 0.15 | 2.64 | 3.06 | 1.43 × 10^{−3} | 1.1 × 10^{−3} | 0.62 | 0.57 |

R10H05s3G30c4 | 0.049 | 0.042 | 1.036 | 1.040 | 1.00 | 4 | 0.30 | 1.46 | 1.59 | 6.87× 10^{−4} | 9.7× 10^{−4} | 0.29 | 0.27 |

R10H05s3G30c2 | 0.057 | 0.046 | 1.008 | 1.040 | 1.00 | 2 | 0.3 | 2.63 | 3.04 | 1.50 × 10^{−3} | 1.1 × 10^{−3} | 0.63 | 0.58 |

μ (Num/Lab) | 0.87 | 1.02 | - | - | - | 1.11 | 0.94 | 0.96 | |||||

σ_{%} (Num/Lab) | 8.0% | 8.4% | - | - | - | 9.7% | 28% | 16% |

**Table 5.**Values of the coefficients c

_{h}and c

_{u}adopted by the several authors in the formulae for the evaluation of h

_{2%}and u

_{2%}based on the scheme of Equations (4) and (5).

Author(s) | Flow Characteristic | Coefficient | Adopted Value/Formulation | Validity Range |
---|---|---|---|---|

Schüttrumpf (2001) [4] | h_{2%}(x_{c} = 0) | c_{h} | 0.33 | cot(α_{off}) = 3, 4 and 6; R_{c} ≥ 0 |

u_{2%}(x_{c} = 0)(flow velocity) | c_{u} | 1.37 | ||

Van Gent (2002) [5] | h_{2%}(x_{c} = 0) | c_{h} | 0.15 | cot(α_{off}) = 4; R_{c} > 0 |

u_{2%}(x_{c} = 0)(flow velocity) | c_{u} | 1.33 | ||

Bosman et al. (2008) [8] | h_{2%}(x_{c} = 0) | c_{h} | 0.01/sin^{2}(α_{off}),for cot(α _{off}) = 4 or 6 | cot(α_{off}) = 4 and 6; R_{c} > 0 |

u_{2%}(x_{c} = 0)(wave celerity) | c_{u} | 0.30/sin(α_{off}),for cot(α _{of}_{f}) = 4 or 6 | ||

Van der Meer et al. (2010) [1] | u_{2%}(x_{c} = 0)(wave celerity) | c_{u} | 0.35∙cot(α_{off}),for cot(α _{off}) = 4 or 6 | cot(α_{off}) = 3, 4 and 6; R_{c} ≥ 0 |

**Table 6.**Error indexes R

^{2}and σ

_{%}characterizing the agreement between the data of h

_{2%}(x

_{c}= 0) and u

_{2%}(x

_{c}= 0) and the new fitting by Equations (5) and (6).

Dataset | h_{2%}(x_{c} = 0)-Equation (5) | u_{2%}(x_{c} = 0)-Equation (6) | h_{2%}(x_{c} = 0)-Equation (3), [8] | u_{2%}(x_{c} = 0)-Equation (4), [8] | ||||
---|---|---|---|---|---|---|---|---|

σ_{%} | R^{2} | σ_{%} | R^{2} | σ_{%} | R^{2} | σ_{%} | R^{2} | |

New laboratory data | 41% | 0.82 | 15.2% | 0.82 | 37.2% | - | 24.8% | - |

New numerical data | 4.1% | 0.99 | 22.1% | 0.73 | 41.9% | - | 71.9% | - |

FlowDike1 data—c3 | 43.4% | 0.85 | 41.8% | 0.79 | 42.7% | 0.85 | 43.2% | 0.77 |

FlowDike2 data—c6 | 42.3% | 0.84 | 50.7% | 0.88 | 43.8% | 0.67 | 69.9% | 0.85 |

© 2019 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**

Formentin, S.M.; Gaeta, M.G.; Palma, G.; Zanuttigh, B.; Guerrero, M.
Flow Depths and Velocities across a Smooth Dike Crest. *Water* **2019**, *11*, 2197.
https://doi.org/10.3390/w11102197

**AMA Style**

Formentin SM, Gaeta MG, Palma G, Zanuttigh B, Guerrero M.
Flow Depths and Velocities across a Smooth Dike Crest. *Water*. 2019; 11(10):2197.
https://doi.org/10.3390/w11102197

**Chicago/Turabian Style**

Formentin, Sara Mizar, Maria Gabriella Gaeta, Giuseppina Palma, Barbara Zanuttigh, and Massimo Guerrero.
2019. "Flow Depths and Velocities across a Smooth Dike Crest" *Water* 11, no. 10: 2197.
https://doi.org/10.3390/w11102197