# Experimental Study of Wave Overtopping Flow Behavior on Composite Breakwater

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Literature Review

#### 2.2. New Wave Overtopping Estimator

#### 2.3. Setup of the Experiments

#### 2.4. Measurement Instruments

#### 2.5. Test Conditions

## 3. Results

#### 3.1. Flow Pattern of the Overtopping Flow

#### 3.2. Overtopping Flow Velocity

#### 3.3. Overtopping Layer Thickness

#### 3.4. Maximum Instantaneous Overtopping Discharge

#### 3.5. Plunging Distance

## 4. Discussion

- Define the following irregular wave parameters for the design structure: the significant wave height (${H}_{s}$), the averaged wave period (${T}_{m}$), and the relative crest freeboard parameter (${R}_{c}$).
- Calculate the wave steepness ($2\pi H/{gT}^{2}$) and the relative crest freeboard parameter (${R}_{c}/H$).
- Compute the empirical parameters, ${a}_{u}$, ${n}_{u}$, and ${b}_{u}$, for Equation (14) and calculate the empirical parameters, ${a}_{h}$, ${n}_{h}$, and ${b}_{h}$, for Equation (15).
- Estimate the expected maximum OFV using Equation (14) and the maximum OLT using Equation (15).
- Plot the OFV and OLT for a given range of wave steepness values and relative crest freeboard.
- Utilize the range of the OFV and OLT for an early investigation of wave overtopping parameters based on the given wave and structure conditions.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

${a}_{h}$ | coefficient in Equation (17) |

${a}_{u}$ | coefficient in Equation (14) |

$B$ | crest width |

${b}_{h}$ | coefficient in Equation (17) |

${b}_{u}$ | coefficient in Equation (14) |

$c$ | wave celerity |

$C$ | coefficient in Equation (8) |

${c}_{A,h}$ | coefficient in Equation (3) |

${c}_{A,u}$ | coefficient in Equation (2) |

${c}_{c,h}$ | coefficient in Equation (6) |

$D$ | coefficient in Equation (9) |

${d}_{p}$ | plunging distance |

$g$ | acceleration due to gravity (9.81 m/s^{2}) |

$H$ | regular wave height |

$h$ | still water depth at structure’s toe |

${H}_{m0}$ | significant wave height from spectral analysis |

${H}_{s}$ | significant wave height defined as highest one-third of wave heights (${H}_{1/3}$) |

${H}_{1/x}$ | average of highest 1/x th of wave heights |

${H}_{rms}$ | root mean square wave height |

${h}_{t}$ | flow thickness |

${h}_{t}\left({x}_{c}\right)$ | wave overtopping layer thickness |

${h}_{t}\left({z}_{A}\right)$ | wave run-up layer thickness |

$K$ | coefficient in Equation (7) |

$L$ | wave length |

$m$ | foreshore slope |

${n}_{h}$ | exponential coefficient in Equation (17) |

${n}_{u}$ | exponential coefficient in Equation (14) |

$Q$ | instantaneous maximum wave overtopping discharge |

${R}^{2}$ | coefficient of determination |

${R}_{c}$ | crest freeboard |

${R}_{u}$ | wave run-up height |

$T$ | regular wave period |

${T}_{p}$ | spectral peak wave period |

${T}_{s}$ | significant wave period (${T}_{H1/3}$) |

${T}_{m-\mathrm{1,0}}$ | spectral wave period |

${T}_{m}$ | averaged wave period |

$u$ | flow velocity |

$u\left({x}_{c}\right)$ | wave overtopping flow velocity |

$u\left({z}_{A}\right)$ | wave run-up velocity |

${x}_{c}$ | location on structure crest, measured horizontally from structure seaward edge |

${z}_{A}$ | location on seaward slope, measured vertically from still water level |

$\alpha $ | angle between structure slope and horizontal |

${\gamma}_{b}$ | influence factor for a berm |

${\gamma}_{B}$ | influence factor for oblique wave attack |

${\gamma}_{f}$ | influence factor for the permeability and roughness of or on the slope |

$\xi $ | Iribarren’s number |

$\pi $ | ratio of a circle’s circumference to its diameter (3.14) |

## Appendix A

Author | Equations | Note |
---|---|---|

EurOtop [5] | Fictitious wave run-up height: $\frac{{R}_{u}}{H}=1.65\xb7{\gamma}_{b}\xb7{\gamma}_{f}\xb7{\gamma}_{\beta}\xb7\xi $ with a maximum of $\frac{{R}_{u}}{H}=1.00\xb7{\gamma}_{fsurging}\xb7{\gamma}_{\beta}\left(4.0-\frac{1.5}{\sqrt{{\gamma}_{b}\xb7\xi}}\right)$ where ${\gamma}_{fsurging}$ is given by ${y}_{fsurging}={\gamma}_{f}+\left(\xi -1.8\right)\xb7\left(1-{\gamma}_{f}\right)/8.2$ Wave run-up velocity at seaward edge: $u({z}_{A}={R}_{c})={c}_{A,u}\left(\sqrt{g\left({R}_{u}-{R}_{c}\right)}\right)$ OFV along the structure crest: $\frac{u\left({x}_{c}\right)}{u\left({z}_{A}={R}_{c}\right)}=\mathrm{exp}\left(-1.4\frac{{x}_{c}}{L}\right)$ | Originally developed for impermeable structure Seadike; however, it is applicable for rough and permeable slope structure. The roughness factor ${\gamma}_{f}$ and empirical coefficient ${c}_{A,u}$ need to be calibrated for different types of armor units and slope angles, as shown in [17,31]. |

Mares-Nassare et al. [20] | $\frac{u}{H/T}={D}_{1}+{D}_{2}m+{D}_{3}\left(\frac{{R}_{c}}{H}-1\right)+{D}_{4}{\xi}^{2}+{D}_{5}\frac{h}{H}\ge 0$ | The equation was derived for rubble mound breakwater with three different armors: 1-layer Cubipod, 2-layer cube and 2-layer rock. The value of coefficient, ${D}_{1}-{D}_{5}$, for each armor unit is available in Table 3. |

Present Study | $\frac{u}{\sqrt{gH}}={a}_{u}{\left(\frac{H}{g{T}^{2}}\right)}^{{n}_{u}}+{b}_{u}\frac{{R}_{c}}{H}$ | The equation was derived based on composite breakwater with 2-layer tetrapod armor unit and permeable core. The value of each coefficient is ${a}_{u}=$ 0.98, ${n}_{u}=-$0.14, and ${b}_{u}=-$1.41. |

Authors | Equations | Note |
---|---|---|

EurOtop [5] | Fictitious wave run-up height: $\frac{{R}_{u}}{H}=1.65\xb7{\gamma}_{b}\xb7{\gamma}_{f}\xb7{\gamma}_{\beta}\xb7\xi $ with a maximum of $\frac{{R}_{u}}{H}=1.00\xb7{\gamma}_{fsurging}\xb7{\gamma}_{\beta}\left(4.0-\frac{1.5}{\sqrt{{\gamma}_{b}\xb7\xi}}\right)$ where ${\gamma}_{fsurging}$ is given by ${y}_{fsurging}={\gamma}_{f}+\left(\xi -1.8\right)\xb7\left(1-{\gamma}_{f}\right)/8.2$ Wave run-up layer thickness at seaward edge: ${h}_{t}({z}_{A}={R}_{c})={c}_{A,h}\left({R}_{u}-{R}_{c}\right)$ OLT along the structure crest: ${h}_{t}\left({x}_{c}\gg 0\right)=\frac{2}{3}{h}_{t}({z}_{A}={R}_{c})$ | The same fictitious wave run-up height equation can be used in OLT estimation as well. It is also recommended to calibrate the roughness factor ${\gamma}_{f}$ and empirical coefficient ${c}_{A,u}$ for different type of armor unit and slope angle. Example of the application can also be found in [17,31]. |

Mares-Nassare et al. [20] | $\frac{{h}_{t}}{H}={C}_{1}+{C}_{2}m+{C}_{3}\left(\frac{{R}_{c}}{H}-1\right)+{C}_{4}\xi +{C}_{5}\frac{h}{H}\ge 0$ | The equation was derived for rubble mound breakwater with three different armors: 1-layer Cubipod, 2-layer cube, and 2-layer rock. The value of coefficient, ${C}_{1}-{C}_{5}$, for each armor units are available in Table 2. |

Present Study | $\frac{{h}_{t}}{H}={a}_{h}{\left(\frac{H}{g{T}^{2}}\right)}^{{n}_{h}}+{b}_{h}\frac{{R}_{c}}{H}$ | The equation was derived based on composite breakwater with 2-layers tetrapod armor unit and permeable core. The value of each coefficient is ${a}_{h}=$ 0.24, ${n}_{h}=-$0.15, and ${b}_{h}=-$0.52. |

## Appendix B

Author | Training Dataset | Test Dataset | ||
---|---|---|---|---|

${\mathit{R}}^{2}$ | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | ${\mathit{R}}^{2}$ | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | |

This study (Equation (14)) | 0.955 | 0.001 | 0.920 | $-$0.010 |

Adibhusana et al. [31] | 0.735 | $-$0.026 | 0.714 | $-$0.005 |

Mares-Nassare et al. [20]: 2-Layer rock 2-Layer cube 1-Layer Cubipod | $<$0.000 $<$0.000 $<$0.000 | $-$0.704 $-$0.728 $-$0.762 | $<$0.000 $<$0.000 $<$0.000 | $-$0.698 $-$0.727 $-$0.761 |

EurOtop [5] | 0.701 | 0.077 | 0.695 | 0.100 |

Author | Training Dataset | Test Dataset | ||
---|---|---|---|---|

${\mathit{R}}^{2}$ | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | ${\mathit{R}}^{2}$ | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | |

This study (Equation (15)) | 0.815 | 0.051 | 0.701 | 0.095 |

Adibhusana et al. [31] | 0.390 | $-$0.027 | 0.237 | 0.083 |

Mares-Nassare et al. [20]: 2-Layer rock 2-Layer cube 1-Layer Cubipod | $<$0.000 $<$0.000 $<$0.000 | 1.897 1.218 1.025 | $<$0.000 $<$0.000 $<$0.000 | 2.147 1.449 1.242 |

EurOtop [5] | 0.340 | $-$0.014 | 0.282 | 0.094 |

Coefficient | Original | Modified | Difference (%) | ||||||
---|---|---|---|---|---|---|---|---|---|

Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | ||||

Training | Test | Training | Test | Training | Test | ||||

${a}_{u}$ | 1.330 | 0.001 | $-$0.010 | 1.396 | 0.095 | 0.087 | 5 | 9,400 | 970 |

${n}_{u}$ | $-$0.136 | 0.001 | $-$0.010 | $-$0.143 | 0.050 | 0.042 | 5 | 4,900 | 520 |

${b}_{u}$ | $-$1.416 | 0.001 | $-$0.010 | $-$1.487 | $-$0.043 | $-$0.057 | 5 | 4,400 | 470 |

Coefficient | Original | Modified | Difference (%) | ||||||
---|---|---|---|---|---|---|---|---|---|

Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | Value | $\mathit{b}\mathit{i}\mathit{a}\mathit{s}$ | ||||

Training | Test | Training | Test | Training | Test | ||||

${a}_{h}$ | 0.305 | 0.051 | 0.095 | 0.321 | 0.246 | 0.308 | 5 | 382 | 224 |

${n}_{h}$ | $-$0.174 | 0.051 | 0.095 | $-$0.182 | 0.181 | 0.239 | 5 | 255 | 152 |

${b}_{h}$ | $-$0.545 | 0.051 | 0.095 | $-$0.572 | $-$0.092 | $-$0.064 | 5 | 280 | 167 |

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**Figure 4.**Dimensions of the composite breakwater used in this study. The dashed line shows the camera’s field of view (FOV).

**Figure 5.**Velocity maps of the wave overtopping flow: (

**a**) reached the rear edge of the breakwater; (

**b**) plunged into the water surface on the rear side of the breakwater.

**Figure 6.**Time series of $u$ and ${h}_{t}$ at the rear edge of the breakwater. The red marker indicates the maximum $u$ and ${h}_{t}$.

**Figure 7.**Scatter plot of wave overtopping flow velocity versus wave steepness and relative crest freeboard.

**Figure 9.**Scatter plot of wave overtopping layer thickness versus wave steepness and relative crest freeboard.

**Figure 12.**Comparison of the calculated and estimated ${Q}_{max}$ values in the training and testing datasets. The calculated ${Q}_{max}$ was obtained using the measured $u$ and ${h}_{t}$ values, while the estimated ${Q}_{max}$ value was obtained by estimating $u$ and ${h}_{t}$ using Equations (14) and (15), respectively.

**Figure 13.**A comparison of the measured and estimated plunging distances (${d}_{p}$) using Equation (18) in both the training and testing datasets.

Authors | ${\mathit{R}}_{\mathit{c}}/\mathit{H}$ | $\mathit{V}/\mathit{H}$ | ${\mathit{c}}_{\mathit{A},\mathit{u}}$ | ${\mathit{c}}_{\mathit{A},\mathit{h}}$ | ${\mathit{c}}_{\mathit{c},\mathit{h}}$ |
---|---|---|---|---|---|

van Gent [23] | 0.7–2.2 | 1/4 | 1.30 | 0.15 | 0.40 |

Schüttrumpf et al. [33] | 0.0–4.4 | 1/3, 1/4, 1/6 | 1.37 | 0.33 | 0.89 |

van der Meer et al. [15] | 0.7–2.9 | 1/3 | 0.35cot$\alpha $ | 0.19 | 0.13 |

EurOtop [5] | - | - | 1.4, 1.5 | 0.20, 0.30 | – |

Mares-Nasarre et al. [17] | 0.34–1.75 | 2/3 | – | 0.52 | 0.89 |

Adibhusana et al. [31] | 0.6–1 | 1/1.5 | 1.21 | 0.21 | – |

Armor Layer | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ | ${\mathit{C}}_{5}$ |
---|---|---|---|---|---|

Cubipod-1L | 0 | $-$4 | $-$1/3 | 0.095 | $-$0.03 |

Cube-2L | 0 | $-$2 | $-$0.3 | 0.085 | $-$0.02 |

Rock-2L | 1/3 | $-$10 | $-$0.45 | 0.080 | $-$0.03 |

Armor Layer | ${\mathit{D}}_{1}$ | ${\mathit{D}}_{2}$ | ${\mathit{D}}_{3}$ | ${\mathit{D}}_{4}$ | ${\mathit{D}}_{5}$ |
---|---|---|---|---|---|

Cubipod-1L | 2 | 20 | $-$2 | 0.20 | $-$1 |

Cube-2L | 4 | $-$30 | $-$2 | 0.20 | $-$1 |

Rock-2L | 2 | $-$30 | $-$2 | 0.20 | $-$0.5 |

Description | Parameter | Ranges |
---|---|---|

Wave period | $T$ | 1.5–3.0 s |

Wave height | $H$ | 0.12–0.20 m |

Relative water depth | $h/L$ | 0.028–0.114 |

Wave steepness | $H/L$ | 0.008–0.057 |

Iribarren’s number | $\xi $ | 2.795–7.216 |

Relative crest freeboard | ${R}_{c}/H$ | 0.6–1 |

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## Share and Cite

**MDPI and ACS Style**

Adibhusana, M.N.; Lee, J.-I.; Ryu, Y.
Experimental Study of Wave Overtopping Flow Behavior on Composite Breakwater. *Water* **2023**, *15*, 4239.
https://doi.org/10.3390/w15244239

**AMA Style**

Adibhusana MN, Lee J-I, Ryu Y.
Experimental Study of Wave Overtopping Flow Behavior on Composite Breakwater. *Water*. 2023; 15(24):4239.
https://doi.org/10.3390/w15244239

**Chicago/Turabian Style**

Adibhusana, Made Narayana, Jong-In Lee, and Yonguk Ryu.
2023. "Experimental Study of Wave Overtopping Flow Behavior on Composite Breakwater" *Water* 15, no. 24: 4239.
https://doi.org/10.3390/w15244239