# Overtopping Metrics and Coastal Safety: A Case of Study from the Catalan Coast

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## Abstract

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## 1. Introduction

## 2. Overtopping Flow Parameters and People Safety

_{max}. Plenty of semi-empirical models have been derived that allow quantifying mean discharge values and depend on the kind of structure and hydraulic boundary conditions. Dikes with gentle slope, namely between 1:7 and 1:2, are studied in [3,4,6,15,16]. Steeper slopes up to vertical walls were analyzed in [17,18,19,20], where the last one includes cases of structures with emergent toe. Goda [21] proposed a set of unified formulas for smooth impermeable sea dikes where new coefficients were derived as a function of local water depth, foreshore slope and dike slope. The influence of the foreshore slope for very and extremely shallow water conditions [22] was taken into account in [15] by means of the equivalent slope concept. Similar to this, [20] employed an imaginary slope for wave run-up and overtopping calculations. Other authors focused their attention on the characterization of overtopping flow depths and velocities, such as [9,10,23,24].

_{f}) is bigger than the maximum available bottom friction resistance of the subject (F

_{r}). The second mechanism models the falling process arising when the unbalancing moment produced by the flow around the feet of the subject is bigger than the restoring moment produced by the weight of the person (M

_{r}). Sandoval and Bruce [12] revisited the model of [11], accounting for the buoyancy of the subject, as well as its related position respect to the incoming flows. The analysis for each mechanism of instability can be derived as follows:

_{f}-F

_{r})

_{r}-M

_{f})

^{3});

_{1}= average diameter of the subject legs (m);

_{d}= drag coefficient (-)

_{1}= distance from pivot point to the center gravity (m);

_{g}= subject’s mass (kg)

_{f}) can be calculated as the drag force applied at the half of the depth. On the other hand, the restoring moment (M

_{r}) is a function of the person’s weight and the distance to the pivot point (d

_{1}).

_{crP}is the critical flow depth, H

_{P}is the height of the subject, while their ratio represents the relative submergence. If H/H

_{crP}< 1, the person is stable. The Froude number is calculated as u⁄√(gd).

## 3. Overtopping Flow Velocity and Flow Depth Estimation on the Dike Crest

_{2%}is the overtopping flow depth on the dike crest, u

_{2%}is the overtopping flow velocity on the dike crest, x

_{c}is the streamwise coordinate on the dike crest, μ is the bottom friction coefficient, Ru

_{2%}is the wave run-up, R

_{c}is the crest freeboard respect to the still water level (Figure 1). The subscript 2% refers to quantities exceeded by 2% of the number of the incident waves. The coefficients c

_{d2%}, c

_{u2%}, c

_{c,d}, and c

_{c,u}are empirical coefficients, and the values can vary according to the literature. For run-up assessment, formulas are here omitted for sake of simplicity. The reader can refer, for example, to [6].

## 4. Case of Study

#### 4.1. Site Description and Model Geometry

#### 4.2. Experimental Setup and Wave Conditions

_{toe}/H

_{m0,deep}being between 0.1 and 0.38, where h

_{toe}is the water depth at the dike toe and H

_{m0,deep}is the deep-water wave height.

_{tot}(L/m); mean overtopping discharge, q (L/s/m), obtained as V

_{tot}/T

_{mm-1,0,deep}*N

_{w,deep}, where T

_{mm-1,0,deep}is the spectral wave period close to the wave generation (offshore) and N

_{w,deep}is the number of waves offshore; maximum overtopping volume, V

_{max}(L/m); flow depth associated to the maximum overtopping volume, d (m); and horizontal velocity associated to the maximum overtopping volume, u (m/s).

#### 4.3. Measurement Setup

_{AWG0}, has been calculated resolving the following system, considering momentum conservation:

_{AWGtmax}is the average velocity as previously described, d

_{AWG0}and d

_{AWG1}are the maximum flow depth values at AWG0 and AWG1 location, respectively, and u

_{AWG0}and u

_{AWG1}are the instantaneous velocities associated to d

_{AWG0}and d

_{AWG1}, respectively. For all analyses reported in the next sections, we will refer to d

_{AWG0}and d

_{AWG1}as maximum flow depth d and overtopping flow velocity u, respectively, for sake of simplicity.

^{2}between 52% and 73% was calculated.

#### 4.4. Wave Analysis

_{m-1,0}. This wave period is shown to be important for many wave-structure interaction processes, and can be used to assess the response of coastal structures with shallow foreshores [15,20]. Hence, the dike was removed, and horizontal bottom followed by absorption material was placed, instead, to measure incident wave conditions at the dike toe, which are required for the analysis. Sensor WG5 was moved to the dike toe location and used for the scope.

#### 4.5. Scale Effects

_{eq}and W

_{eq}) were calculated. The results were compared versus the proposed critical limits, namely R

_{eq}> 10

^{3}and W

_{eq}> 10. In total, 25 cases showed a R

_{eq}< 10

^{3}and W

_{eq}< 10. Further analysis was carried out to quantify scale effects on those cases following two different methodologies: (1) a correction for the model scale, based on [6] was calculated; (2) an artificial neural network (ANN) proposed by [31] was employed, and the predicted overtopping discharges were compared with the measured ones. Both methodologies prove that scale effects can be neglected. The correction calculated with [6] method ranged between 1 and 2.5. ANN predictions show values in the same order or just smaller than experimental ones. Further details are here omitted for the sake of simplicity.

## 5. Results

#### 5.1. Relationship between Mean Discharge, Individual Volumes and Overtopping Flow Parameters

_{max}seems pretty linear, whereas more dispersion can be noticed looking at the overtopping flow depth and velocity behavior versus individual overtopping volume (Figure 11). This is especially true for the 1:15 foreshore slope, where more intense overtopping events are far more energetic and violent than the ones for 1:30 slopes.

_{m0}and T

_{m-1,0}are the spectral wave height and period at the dike toe, respectively, h

_{toe}is the water depth at the dike toe, B is the width of the dike promenade, g is the gravity acceleration, q is the average overtopping discharge, V

_{max}the maximum individual overtopping volume expressed in L per meter of crest width, R

_{c}the crest freeboard, and u and d are the overtopping flow velocity and depth, respectively. The parameter tanθ

_{eq}corresponds to the equivalent slope, calculated starting from the dike and foreshore slope for cases with foreshores in shallow water conditions, as indicated in [14]. Hence, n = 11 and k = 2, leading to nine dimensionless parameters:

_{toe}, corresponding to the wave celerity in shallow waters. Identification of dimensionless groups will help to investigate possible relationships between overtopping flow depth and velocity with other variables at stake. The relationship between overtopping flow depth and velocity with individual maximum overtopping volumes, has been investigated in terms of dimensionless groups as shown in Figure 14. It is possible to distinguish two different trends, one per foreshore slope. Lower values of the dimensionless velocity are shown for a wide range of volumes in case of the 1:30 slope. Opposite to that, a wide variation of velocities is shown within a relatively short range of volumes for the steeper foreshore. A more careful analysis of the results shows that the overtopping flow depth is greatly affected by the promenade width (see Figure 15), which does not show a clear correlation with overtopping flow velocity.

_{2%}or R

_{max}. The results are depicted in Figure 16, where it is clear that poor agreement is found between the experimental results and calculated values.

#### 5.2. Overtopping Flow Parameters Expressed in Terms of Individual Maximum Overtopping Volume

_{m-1,0}can be found at the denominator of Equations (6)–(8); however, the measured wave periods are far larger than the one tested in [24], as deeply affected by heavy breaking and release and shift of the spectral energy to very low frequency as result of the release of infra-gravity waves for very and extremely shallow water conditions.

_{c}, where flow parameters are measured, of the surf similarity parameter ξ

_{m-1,0}and dimensionless freeboard R

_{c}/H

_{m0}. The surf similarity parameter is here calculated by employing the equivalent slope as defined in [15]. The influence of the freeboard might be explained as follows: for the same volume, a higher freeboard will lead to lower flow depths (run-up is lower). In addition, the surf similarity parameter will be bigger for longer periods (i.e., bigger wavelengths), leading to bigger individual discharges.

^{2}is about 50%). In any case, it is important to emphasize here that it is not intended to find new relationships for the overtopping flow parameters to overcome or upgrade the ones already proposed in the literature. The EPR analysis is carried out to provide a general overview of the possible correlations among variables and help to interpret the results. Larger databases are required to optimize any regression for the flow properties on the dike crest, but this is out of scope of the present work.

## 6. Analysis and Comparison of Safety Criteria and Limits of Wave Overtopping for Design of Sea Dikes

_{max}< 1,000; 1,000 < V

_{max}< 5000; V

_{max}> 5000 L/m). Stability curves calculated for a male adult person and a 10 year old child are plotted in Figure 19 and Figure 21, respectively. All data above the safety curves lie in an unsafe region, whereas all data below the lines correspond to safe flow conditions.

_{max}> 1,000 L/m fall within the unsafe region (above the curves in the figures), while the same volume can be not so critical for longer berms, for which the results are more scattered. Cases with V

_{max}> 1,000 L/m that lead to unsafe flows are those with mean discharge q > 5/L/s/m, apart from a few exceptions. In general, very high discharges and volumes, respectively greater than 10 L/s/m and 5000 L/m, lead to unsafe flows for both considered promenade widths.

## 7. Conclusions

- Tolerable discharge values proposed by [6] vary depending on the local wave height at the toe of the coastal structure. On the contrary, a fixed value corresponding to 600 L/m is reported as a threshold for individual overtopping volume. It is not clear from [6] whether this value corresponds to some specific value of overtopping flow velocity as, for example, in [5]. If one criterion is fulfilled, it can happen that the other criteria appear stronger. For the case study, average discharges were always within the proposed limits, whereas individual volumes were above the tolerable value.
- Overtopping flow velocities and depths are plotted along with the corresponding maximum volumes and average discharges. What emerges is a not clear two-way relationship between maximum overtopping volumes and velocities or flow depth: a dependence does exist, as also confirmed by applying the EPR technique [33] to the present dataset. Nevertheless, the data scatter is big, and therefore a larger dataset is required to performed more detailed regression analysis on the data.
- Experimental values of overtopping flow velocities and flow depth have been compared with stability curves for pedestrians (adults and children) placed on the sea dike and subjected to overtopping waves. The results show a clear influence of the dike crest width, where for mean discharges lower than 5 L/s/m and volumes lower than 1,000 L/m, a shorter crest does not necessarily lead to safe conditions, where the longer crest shows a combination of values of overtopping flow parameters lower than the thresholds calculated using [12,13].
- Volumes bigger than 600 L/m do not always determine unsafe conditions for pedestrians. At least 20% of all analyzed data are in the safe region, for the specific case of study.
- EurOtop [6] tolerable limits and stability curves lead to discordant results. In fact, due to the non-two-way relationship between volumes and corresponding flow parameters, it can be observed that flow parameters related to 1,000 L/m maximum volumes can be located in the unsafe area, while the same parameters related to bigger volumes can even be included in the safety range for a large enough crest.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Disclaimer

## References

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**Figure 4.**Bathymetry extracted from the LIM/UPC drone survey (

**left plot**) and resulting profiles by merging existing data with the carried-out survey (

**right plot**). A photo of the drone is in the small picture.

**Figure 5.**CIEMito wave flume—drawing of the longitudinal section (distorted). Dimensions are in model scale. The value of m is equal to 15 and 30, respectively. Accordingly, n assumes values of 8 and 5, respectively.

**Figure 8.**Correlation between overtopping flow depths measured with ultrasonic sensors and high-speed cameras.

**Figure 9.**Example of wave spectrum for incident wave conditions at the dike toe (values in model scale).

**Figure 10.**Variation of maximum individual overtopping volume with mean discharge (dimensions in prototype). Limits for safety of pedestrians and assets based on EurOtop (2018) are shown. The two vertical blue lines correspond to tolerable discharges of 10 L/s/m and 20 L/s/m respectively, for H

_{m0}= 1 m.

**Figure 11.**Variation of overtopping flow velocity and depth with maximum individual overtopping volume (dimensions in prototype) for two different foreshore slopes.

**Figure 12.**Variation of overtopping layer velocity and depth with average overtopping discharge (dimensions in prototype) for two different foreshore slopes.

**Figure 13.**Variation of overtopping layer velocity and depth with deep-water wave characteristics (dimensions in prototype) for two different foreshore slopes.

**Figure 14.**Variation of dimensionless overtopping layer velocity and depth with dimensionless maximum individual overtopping volume for two different foreshore slopes.

**Figure 15.**Variation of dimensionless overtopping layer velocity and depth with dimensionless maximum individual overtopping volume for two different promenade width.

**Figure 16.**Estimation of overtopping flow depth and velocity employing existing formulas from Schüttrumpf and Van Gent (2003), Trung (2014) and EurOtop (2018).

**Figure 19.**Flow depth versus velocity, comparison with Sandoval and Bruce (2017) curves, the discharges are divided for different promenades and measured average discharge.

**Figure 20.**Froude number versus relative submergence, comparison with Arrighi (2017) curve, the discharges are divided for different promenades and measured average discharge.

**Figure 21.**Flow depth versus velocity, comparison with Sandoval and Bruce (2017) curves, the discharges are divided for different promenades and maximum volumes.

**Figure 22.**Froude number versus relative submergence, comparison with Arrighi (2017) curve, the discharges are divided for different promenades and maximum volumes.

Hazard Type and Reason | Mean Discharge, q (L/s/m) | Maximum Individual Volume V_{max} (L/m) |
---|---|---|

People at structures with possible violent overtopping, mostly vertical structures | No access for any predicted overtopping | No access for any predicted overtopping |

People at seawall/dike crest. Clear view of the sea. | - | - |

H_{m0} = 3 m | 0.3 | 600 |

H_{m0} = 2 m | 1 | 600 |

H_{m0} = 1 m | 10–20 | 600 |

H_{m0} < 0.5 m | No limit | No limit |

Scale | h_{toe} (m) | R_{c} (m) | Promenade Width (m) |
---|---|---|---|

Model | 0.009–0.019–0.024–0.029 | 0.81–0.071–0.066–0.061 | 0.12–0.24 |

Prototype | 0.45–0.905–1.2–1.45 | 4.05–3.55–3.3–3.05 | 6–12 |

Sensor | WG0 | WG1 | WG2 | WG3 | WG4 | WG6 | WG7 | WG5 |
---|---|---|---|---|---|---|---|---|

x (m = 15) | 2.8 | 2.96 | 3.15 | 3.40 | 3.69 | 7.20 | 8.20 | 9.20 |

x (m = 30) | 2.8 | 2.96 | 3.15 | 3.40 | 3.69 | 7.20 | 8.20 | 9.20 |

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**MDPI and ACS Style**

Altomare, C.; Gironella, X.; Suzuki, T.; Viccione, G.; Saponieri, A. Overtopping Metrics and Coastal Safety: A Case of Study from the Catalan Coast. *J. Mar. Sci. Eng.* **2020**, *8*, 556.
https://doi.org/10.3390/jmse8080556

**AMA Style**

Altomare C, Gironella X, Suzuki T, Viccione G, Saponieri A. Overtopping Metrics and Coastal Safety: A Case of Study from the Catalan Coast. *Journal of Marine Science and Engineering*. 2020; 8(8):556.
https://doi.org/10.3390/jmse8080556

**Chicago/Turabian Style**

Altomare, Corrado, Xavi Gironella, Tomohiro Suzuki, Giacomo Viccione, and Alessandra Saponieri. 2020. "Overtopping Metrics and Coastal Safety: A Case of Study from the Catalan Coast" *Journal of Marine Science and Engineering* 8, no. 8: 556.
https://doi.org/10.3390/jmse8080556