# Reduction of Wave Overtopping and Force Impact at Harbor Quays Due to Very Oblique Waves

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Overtopping and Force Reduction

#### 2.1. Vertical Quay

^{3}/s/m), H

_{m0}is the significant incident wave height, measured at the toe of the structure (m), R

_{c}is the crest freeboard (m), and γ

_{β}is the reduction coefficient that considers the effects of the obliqueness (-).

_{β}is expressed in EurOtop as

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

_{s,toe}is the incident wave height at the toe of the dike. The coefficients b

_{1}, c

_{1}, b

_{2}, and c

_{2}depend on the foreshore slope as summarized in Table 1.

#### 2.2. Sloping Dike

_{β}is expressed as

_{β}is identical as in (7).

_{f}has been assumed equal to 1 (smooth slope). Van Doorslaer et al. [9] propose a reduction factor γ

_{prom_v}to take into account the presence of a storm return wall on the top of the dike. This coefficient considers both the effect of the wall height and position. The values of γ

_{prom_v}are calculated for each case based on the approach described in Van Doorslaer et al. [9].

_{m−1,0}is the spectral wave length calculated using the spectral period in deep waters T

_{m−1,0}= m

_{−1}/m

_{0}.

_{wall}) and freeboard (R

_{c}) as follows:

#### 2.3. Force Reduction

_{c}from the still water level to the top of the storm wall of 3.18 m, 2.22 m, and 1.20 m (prototype values). The irregular waves had a Jonswap wave spectrum (γ = 3.3). The significant wave height H

_{m0}ranged from 0.78 m to 3.00 m (prototype values); the wave period T

_{p}was either 7.00 s or 10.00 s. The experiments were carried out in two dimensional conditions with perpendicular waves (no wave obliqueness). The authors proposed a new formula to evaluate the wave force on a storm wall, both for quay walls and sea dikes. The formula can be expressed as follows:

_{1/250}is the average force of the highest 1/250 waves. The coefficients a and b (Table 2) are derived from a non-linear regression analysis and they are considered as the mean value of normally distributed variables. Under this hypothesis, the relative standard deviation (σ’ = σ/μ) was calculated for each coefficient and is reported in Table 2 between brackets.

## 3. Methods and Instrumentation

#### 3.1. Model Settings of Overtopping Tests

#### 3.1.1. Instrumentation

#### 3.1.2. Test Programme

_{c}) can assume negative (i.e., Still Water Level, SWL, above the dike crest) and positive values (SWL below the dike crest). Three different wall heights were used (0 m, 1 m, and 2 m in prototype scale). The wall elevation with respect to the SWL defines the dike freeboard R

_{c}. The berm (distance between the storm return wall and the quay or dike crest) lengths used in the experiments were 0, 5, 25, and 50 m in prototype scale. Tests with no reliable measured wave conditions, zero overtopping, and water volumes exceeding the boxes’ volume, as well as preliminary tests to set-up the model were excluded from further analyses.

- -
- For each test the berm length was calculated as a distance between the edge (crest) of the quay (sea dike) and the crown wall.
- -
- For each angle the projection of the berm length was measured on the wave direction; this is the effective berm length that the wave has to run before reaching the wall.
- -
- To calculate the mean overtopping for the entire quay some buffer zones at both edges of the structure were skipped (where possible model effects are noticed). For instance, in the case with no crown wall or crown wall on the quay edge, the entire quay length (8 m) was considered excluding the two overtopping boxes situated at the edges of the structure.
- -
- It was verified on video recordings that the peaks in the overtopping volume were not due to model effects (boundary reflection), but they were due to the wave attack.

#### 3.2. Model Settings of Force Test

#### Instrumentation

## 4. Results

#### 4.1. Overtopping Reduction

- For each test, the berm length was calculated as a distance between the edge of the quay (sea dike) and the crown wall.
- For each angle, the projection of the berm length along the wave direction was assessed; this represents the effective berm length that the wave has to run before reaching the wall.
- Starting from the first corner of the dike, the projection of the effective berm along the quay gives the minimum distance before which no wave reaches the wall.
- The width considered to calculate the mean overtopping for the entire quay is equal to the quay length minus the calculated distance and some buffer zones at the edge (where possible model effects are noticed).

- -
- Sloping dike: only CLASH data with slope between 1:4 and 1:2 with gentle or no foreshore were considered.
- -
- Vertical quay: only tests with gentle or without foreshore were considered.

#### 4.1.1. Vertical Quay Wall

_{w}, equal to 0 m) and with a crest berm (d

_{w}larger than 0 m). The dash–dot lines indicate a prediction of 10 times larger and smaller with respect to the central line (ratio predicted/measured equal to 1:1). The formula overestimates the overtopping discharge for the 70° and 80° directions, while for the 0°, 45°, and 60° directions results are in reasonable agreement or within the above mentioned range.

- (a)
- the correction coefficient represents an upper limit (safe approach) for the present cases with very oblique waves, although not excessively high as EurOtop [1]; and
- (b)
- the expression for γ
_{β}is applicable up to 80°, meanwhile EurOtop [1] indicates a constant value for wave angles larger than 45°.

- A = 0.040 and B = 2.6 in EurOtop [1],
- A = 0.033 and B = 2.3 in Goda [3], and
- A = 0.116 and B = 3.0 in Franco and Franco [18].

_{β}is a function of the A and B coefficients. The differences between Goda [3] and EurOtop [1] can be considered negligible because the values of A and B coefficients are rather similar.

- γ
_{β}= 0.76 (σ = 0.23), for β = 45°; - γ
_{β}= 0.75 (σ = 0.17), for β = 60°; - γ
_{β}= 0.44 (σ = 0.21), for β = 70°; and - γ
_{β}= 0.28 (σ = 0.04), for β = 80°.

_{β}estimated for each single test. The confidence interval represented in Figure 10 is calculated as ±σ with respect to the mean value. As general approach, the mean value of γ

_{β}has to be used for design purposes. It can be noticed that the difference in the reduction coefficient between 0.72 (calculated value using EurOtop [1]) and 0.28 might cause a difference in the calculated discharge of at least 1 order of magnitude (10 times) in the selected data range.

_{m0}

^{3})

^{^0.5}. Only the FHR cases with the wall on the edge of the quay are plotted in order to avoid misinterpretations due to the effects of the width of the crest berm. Three different plots are shown in Figure 11:

- (a)
- the values of Q are plotted against the non-dimensional freeboard R
_{c}/H_{i}; - (b)
- the values of Q are plotted against the non-dimensional freeboard R
_{c}/H_{i}γ_{β}(EurOtop), where γβ (EurOtop) is the correction coefficient calculated using the EurOtop (2007) formula; and - (c)
- (the values of Q are plotted against the non-dimensional freeboard Rc/Hiγβ(Goda), where γβ (Goda) is the correction coefficient calculated using the Goda [3] formula.

_{p}

^{2}that can be assumed as the wave length in deep water conditions. The combination of obliqueness, wall height and berm length made it challenging to have a clear view of the phenomena occurring at the structure. Despite the rather wide data scatter, there are clear differences between short or no berm layouts and wide berm layouts. A dependence on the berm length can be detected, the overtopping was reduced when the ratio of the berm length over the wave length was increased and this trend was clearer for larger wave angles. The waves travelled at the dike crest before approaching the storm wall and it was expected that the waves would refract on the berm, and therefore approach the wall with less obliqueness, but still not perpendicular. The distance travelled by the waves to reach the wall was larger for larger angles, so the amount of energy dissipated on the crest might have been larger. The configurations without berm, and with short berm length, 5 m in prototype, show a similar behaviour leading to larger overtopping discharge than the configurations with wider berms (25 m and 50 m in prototype).

#### 4.1.2. Sloping Dike

- (a)
- the values of Q are plotted against the non-dimensional freeboard R
_{c}/H_{i}; - (b)
- (c)
- the values of Q are plotted against the non-dimensional freeboard R
_{c}/H_{i}γ_{β(Goda)}γ_{prom_v}, where γ_{β(Goda)}is the correction coefficient calculated using the Goda [3] formula.

#### 4.2. Force Reduction

- (a)
- measured wave force on the storm wall for the quay wall layout;
- (b)
- measured wave force on the storm wall for the sea dike layout;
- (c)
- measured wave force on the storm wall for the quay wall layout, including the correction with the proposed reduction factor for wave obliqueness; and
- (d)
- measured wave force on the storm wall for the sea dike layout, including the correction with the proposed reduction factor for wave obliqueness.

_{1/250}/ρgR

_{c}

^{2}as function of the relative freeboard, both for FHR and UPC results. A common trend between the two experimental datasets can be noticed, despite of a certain scatter in the FHR results, mainly due to the different wave angles.

## 5. Conclusions

- The EurOtop formula [1] generally overestimates the overtopping discharge for large wave obliqueness.
- The values of the reduction factor γ
_{β}calculated for the vertical quay layout are equal to 0.76, 0.75, 0.44, and 0.28, respectively, for 45°, 60°, 70° and 80°. - The values of the reduction factor γ
_{β}calculated for the sloping dike layout are equal to 0.72, 0.54 and 0.44 respectively for 45°, 60° and 80°. - A rather large scatter is present in the results similar to the results presented in previous studies [3].
- The expression of γ
_{β}presented by Goda [3] is finally proposed as a good compromise between accuracy (in comparison with physical model results) and a certain safety in the design of the storm walls. - The high obliqueness combined with long berms on the crest (comparable with the wave length) leads to very low or zero overtopping discharge.
- The berm length (ranging from 0 to 50 m) has a larger influence on the overtopping discharge than the wall height (ranging from 1 to 2 m).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Model set-up based on the real cases from Belgian harbors. Upper part: vertical quay design and an example of storm return wall at Ostend harbor. Lower part: sloping dike design and an example at Blankenberge harbor.

**Figure 3.**The position of the structure in the wave basin during experiments: (

**a**) wave directions 60°, 70° and 80°; (

**b**) perpendicular wave attack; (

**c**) wave direction 45°.

**Figure 8.**Overtopping discharge per box along the vertical quay and the total wave height for direction 60°.

**Figure 9.**Quay wall: predicted [1] vs. measured overtopping discharges. The circles indicate the cases without berm crest (d

_{w}= 0), the triangles indicate the cases where a berm crest is present (d

_{w}> 0).

**Figure 10.**Quay wall: variation of reduction coefficient with wave angle, comparison to existing formulas.

**Figure 11.**CLASH and FHR (wall on the edge of the quay) data vs. EurOtop predictions: (

**a**) overtopping plotted against the non-dimensional freeboard; (

**b**) overtopping plotted against the non-dimensional freeboard, with the correction factor from EurOtop (2007) formula; (

**c**) overtopping plotted against the non-dimensional freeboard, with the correction factor from Goda (2009) formula.

**Figure 12.**Non-dimensional discharge vs. relative wall height and relative berm length (quay layout).

**Figure 14.**Sloping dike: variation of reduction coefficient with wave angle; comparison with existing formulas.

**Figure 16.**Dependence of the wave forces on the incident wave height for different wave obliqueness.

**Figure 18.**Dependence of the wave force on the relative freeboard with and without reduction factor (

**a**and

**c**: quay wall;

**b**and

**d**: dike).

**Figure 19.**Dependence of the non-dimensional wave forces on the relative freeboard and comparison with data from UPC [10].

**Figure 20.**Measured forces versus Van Doorslaer et al. [10] predictions for FHR 0° cases and UPC cases.

**Figure 21.**Application of Van Doorslaer et al. [10] formula with and without reduction factor for wave obliqueness.

**Table 1.**Optimum coefficient values of empirical formulas for intercept A and gradient coefficient B (after Goda, 2009).

Seabed Slope | Coefficient A | Coefficient B | ||||
---|---|---|---|---|---|---|

tan θ | A0 | b1 | c2 | B0 | b2 | c2 |

1/10 | 3.6 | 1.4 | 0.1 | 2.3 | 0.6 | 0.8 |

1/20–1/1000 | 3.6 | 1.0 | 0.6 | 2.3 | 0.8 | 0.6 |

**Table 2.**Coefficients a and b in Equation (13) for different geometries (after Van Doorslaer et al. [10]).

Geometry | a | b |
---|---|---|

Dike | 8.31 (0.22) | 2.45 (0.07) |

Quay | 18.27 (0.23) | 3.99 (0.06) |

All | 5.96 (0.23) | 2.42 (0.09) |

Total no. of Tests | Used for Analyses | Vertical Quay | Sloping Dike (1 to 2.5) | |
---|---|---|---|---|

377 | 230 | 191 | 39 | |

Wave directions | Wave height (H_{m0}) | Wave period (T_{p}) | Crest freeboard (R_{c}) | Storm return wall position |

0°, 45°, 60°, 70°, 80° | 0.96 to 3.39 m | 5.1 to 12.6 s | 0 to 2.75 m | 0 to 50 m |

Total Number of Tests | 44 | |||
---|---|---|---|---|

Wave directions | Wave height (H_{m0}) | Wave period (T_{p}) | Crest freeboard (R_{c}) | Storm return wall position |

0°, 45°, 80° | 1.04 to 4.54 m | 10.2 to 12.9 s | 0 to 3.0 m | 0 to 25 m |

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## Share and Cite

**MDPI and ACS Style**

Dan, S.; Altomare, C.; Suzuki, T.; Spiesschaert, T.; Verwaest, T.
Reduction of Wave Overtopping and Force Impact at Harbor Quays Due to Very Oblique Waves. *J. Mar. Sci. Eng.* **2020**, *8*, 598.
https://doi.org/10.3390/jmse8080598

**AMA Style**

Dan S, Altomare C, Suzuki T, Spiesschaert T, Verwaest T.
Reduction of Wave Overtopping and Force Impact at Harbor Quays Due to Very Oblique Waves. *Journal of Marine Science and Engineering*. 2020; 8(8):598.
https://doi.org/10.3390/jmse8080598

**Chicago/Turabian Style**

Dan, Sebastian, Corrado Altomare, Tomohiro Suzuki, Tim Spiesschaert, and Toon Verwaest.
2020. "Reduction of Wave Overtopping and Force Impact at Harbor Quays Due to Very Oblique Waves" *Journal of Marine Science and Engineering* 8, no. 8: 598.
https://doi.org/10.3390/jmse8080598