# Numerical Study on Wave Dissipation and Mooring Force of a Horizontal Multi-Cylinder Floating Breakwater

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## Abstract

**:**

## 1. Introduction

## 2. Establishment and Verification of Numerical Model

#### 2.1. Establishment of Numerical Model

- (i)
- At the seabed (z = −d),

- (ii)
- For the free surface (z = 0),

^{2};

- (iii)
- For the immersed surface of the object,

_{j}is the velocity potential on the jth degree of freedom; and v

_{i}is the motion speed on the jth degree of freedom.

_{t}:

_{t}= H

_{t}/H

_{i}

_{t}is the transmission wave height, and H

_{i}is the incident wave height. A decreased transmission coefficient signifies a more effective wave attenuation performance.

#### 2.2. Verifications of Numerical Model

#### 2.2.1. Modeling of Double-Box Floating Breakwater

^{6}N/m. A numerical model with three-dimensional characteristics, duplicating the dimensions and wave elements of the experimental model, was developed to effectively simulate the hydrodynamic behavior of double-row floating breakwaters.

#### 2.2.2. Modeling of Multi-Cylinder Floating Breakwater

_{i}= 0.09 m and 0.15 m. The numerical model’s findings exhibit a high level of concurrence with the experimental results. Overall, the numerical outcomes demonstrate consistency with the experimental data, with the discrepancy falling within an acceptable range.

## 3. Comparisons among Floating Breakwaters with Different Cross-Sections

^{2}. The dimensions of each cross-section configuration are as follows: (a) the large cylinder possesses a diameter of 9.6 m, while the two small cylinders have a diameter of 2.4 m each; (b) the section is circular with a diameter of 10.2 m; and (c) the section is square, with each side measuring 9.0 m. Each floating breakwater has a total length of 100 m and is secured by four mooring chains, with two mooring chains positioned upstream and downstream, respectively. The mooring chains are made of steel and possess a rigidity of 2.33 × 10

^{8}N/m, while the spacing between them is 80 m. The lower end of the mooring chain is firmly attached to the sea’s bottom and is positioned 100 m horizontally from the breakwater. The incident wave height ranges from 1.5 to 3.5 m with a period of 6.0 s. The water depth is measured at 50 m, and the wave incidence direction is perpendicular to the long side of the breakwater.

_{i}= 4.0 m and T = 6.0 s, respectively. The combination of the incident wave and the reflected wave leads to an increased wave amplitude in the front of the breakwater, while the wave height behind the breakwater is dissipated to some extent. The wave height in front of the multi-cylinder breakwater (Figure 5c) exhibits a greater magnitude compared to the other two breakwater types owing to its deeper draft. Additionally, wave breaking transpires in the vicinity of smaller cylinders situated closer to the water’s surface. Both of these factors contribute to the lowest wave height behind the multi-cylinder breakwater.

## 4. Analysis of Wave Dissipation Capacity of Multi-Cylinder Floating Breakwater

#### 4.1. Effects of Influential Factors on Wave Transmission Coefficient

_{1}). Figure 8 shows the wave height extracted along the centerline of the breakwater in the perpendicular direction. The wave height is taken from 175 m in front of and 175 m behind the breakwater, with the location of the breakwater at x = 0 m. It can be observed that the wave heights in front of and behind the breakwater have different values. Then, the averaged wave height behind the breakwater is utilized to calculate the wave transmission coefficient.

#### 4.1.1. Effect of Large Cylinder Diameter

_{1}= 5.0 m~12.0 m while other parameters are D

_{2}= 1.8 m, α = 45°, T = 6.0 s, and H

_{i}= 2.0 m and the water depth is d = 50 m. The numerical results of wave clouds around the floating breakwater are shown in Figure 7. The wave elevations along the centerline of the floating breakwater in wave propagation directions are shown in Figure 8. The transmission coefficients are calculated from the wave elevations behind the floating breakwater. As shown in Figure 9a, the transmission coefficient of the large cylinder decreases as its diameter increases due to the increased reflection of wave energy by the larger diameter. Simultaneously, the presence of the breakwater reduces the passage of diffracted waves through its bottom, thereby enhancing the wave dissipation capacity of the large-diameter multi-cylinder floating breakwater.

#### 4.1.2. Effect of Small Cylinder Diameter

_{2}) ranges from 1.0 to 2.0 m. The remaining parameters of the cylinder breakwater, namely D

_{1}= 10.0 m, α = 45°, T = 6.0 s, H

_{i}= 2.0 m, and d = 50 m, remain constant. Figure 9b illustrates the correlation between the transmission coefficient of the breakwater and the diameter of the small cylinder. The transmission coefficient exhibits an overall range of 0.3 to 0.6 and diminishes with an increase in the small cylinder’s diameter. This phenomenon arises due to the unaltered diameter of the large cylinder, whereby an enlarged small cylinder diameter permits a greater passage of wave energy beneath the large cylinder, consequently leading to the increased transmission of wave energy through the breakwater.

#### 4.1.3. Effect of the Angular Position of Small Cylinder

_{2}= 1.8 m, D

_{1}= 10.0 m, T = 6.0 s, H

_{i}= 2.0 m, and d = 50 m. Figure 9c illustrates the correlation between the transmission coefficient and the variation in the angular position between the two small cylinders. It is evident that the transmission coefficient exhibits a distribution ranging from 0.3 to 0.6, and it tends to decrease as the central angle increases. This phenomenon can be attributed to the increased distance between the two small cylinders as the central angle expands, leading to wave interference and disruption between the small cylinders. Simultaneously, there is a gradual decrease in the height of the small cylinders, accompanied by an increase in the draft of the small cylinders. As a result, the angular position changes from 30° to 90°, leading to the gradual strengthening of the breakwater’s wave dissipation capacity.

#### 4.1.4. Effect of Incident Wave Period

_{1}= 10.0 m, D

_{2}= 1.8 m, H

_{i}= 2.0 m, α = 45°, and d = 50 m, remain constant. Figure 9d illustrates the relationship between the transmission coefficient and the incident wave period. The transmission coefficient generally varies between 0.2 and 0.9, indicating that the incident wave period significantly influences the transmission coefficient. Moreover, the transmission coefficient exhibits a gradual increase as the incident wave period increases. The reduction effect of the breakwater on the long wave is less clear than that of the short wave for the following three reasons: (i) the long-period incident wave has the ability to pass beneath the breakwater, resulting in a portion of its energy remaining unconsumed by the breakwater; (ii) when the long-period incident wave encounters the breakwater, it induces a vigorous motion response, causing significant agitation and subsequently elevating the wave height behind the breakwater; (iii) the breakwater’s ability to dissipate long-wave energy is relatively limited compared to its effectiveness for short waves; (iv) the wave overtopping induced by long-period waves surpasses that of short-period waves, thereby facilitating the infiltration of certain waves beyond the breakwater and into the protected region.

#### 4.1.5. Effect of Incident Wave Height

_{i}, was changed from 1.5 m to 3.5 m while keeping the other parameters constant as follows: D

_{1}= 10.0 m, D

_{2}= 1.8 m, α = 45°, and d = 50 m. Figure 9e illustrates the relationship between the transmission coefficient of the breakwater and the incident wave height. The transmission coefficient exhibits a clear range between 0.3 and 0.9, signifying a notable correlation with the incident wave height. As the incident wave height increases, the transmission coefficient gradually rises. This phenomenon can be attributed to the overtopping caused by certain waves surpassing the breakwater and entering the protected area behind it, consequently augmenting the energy of the waves that traverse the breakwater. Consequently, the transmission coefficient of the breakwater exhibits an increase.

#### 4.2. Empirical Formula

_{t}of the multi-cylinder floating breakwater is determined for different influential factors. To establish an empirical formula, the fitting process employs the multivariate linear regression method, resulting in the derivation of the following formula.

_{1}/D

_{2}), the angular position of the smaller cylinders (α/360°), the ratio of the wave period to the natural period (T/T

_{0}), and the ratio of the wave height to the water depth (H

_{i}/d). Consequently, a correlation between the transmission coefficient and the aforementioned parameters is established as follows:

_{1}/D

_{2}≤ 10, 0.13 ≤ α/360° ≤ 0.25, 0.33 ≤ T/T

_{0}≤ 0.67, and 0.30 ≤ H

_{i}/d ≤ 0.70. The correlation coefficient, R = 0.882 (as illustrated in Figure 10), suggests a robust correlation between the transmission coefficient and the four dimensionless parameters, enabling the accurate prediction of the transmission coefficient of the present multi-cylinder floating breakwater.

## 5. Analysis of Total Mooring Force of Multi-Cylinder Floating Breakwater

#### 5.1. Effects of Influential Parameters of Mooring Chain Force

_{i}), incident wave period (T), and water depth (d). The configuration parameters for the multi-cylinder floating breakwater were set as follows: D

_{1}= 12.0 m, D

_{2}= 2.0 m, α = 60°, and were determined based on the minimum wave transmission coefficient according to the local design wave parameter.

#### 5.1.1. Effect of Wind Speed

_{i}= 4.6 m, T = 7.6 s, d = 50 m. The forces on each individual mooring chain are determined using a numerical model, taking into account the specific environmental conditions. The total mooring force is then calculated as the sum of these four mooring chain forces in a time series. Then, the relationship between the maximum total mooring force and the wind speed is obtained, as shown in Figure 11a. Based on the depicted relationship, it can be deduced that the maximum total mooring force exhibits a positive correlation with wind speed, wherein the force increases with the increasing wind speed. Specifically, within the wind speed range of V = 15 m/s to V = 55 m/s, the maximum total mooring force experiences a modest increase from 1.854 × 10

^{7}N to 1.859 × 10

^{7}N.

#### 5.1.2. Effect of Current Velocity

_{i}= 4.6 m, T = 7.6 s, and d = 50 m remain constant. Subsequently, the correlation between the maximum total mooring force of the breakwater and the current velocity is determined based on the numerical results and is visually represented in Figure 11b.

^{7}N to 1.983 × 10

^{7}N. It indicates that the impact of current velocity on the total mooring force surpasses that of wind speed. The presence of the breakwater diminishes the current velocity in the area behind the structure, resulting in enhanced water stability. Consequently, as the current velocity escalates, the mooring force exerted by the breakwater exhibits a positive correlation.

#### 5.1.3. Effect of Incident Wave Height

_{i}) ranges from 1.0 to 6.0 m, while other parameters such as velocity (V = 35 m/s), current velocity (u = 1.0 m/s), wave period (T = 7.6 s), and water depth (d = 50 m) remain constant. Figure 11c displays the relationship curve between the maximum total mooring force and the incident wave height. It can be concluded that there is a positive correlation between the maximum total mooring force and the incident wave height. Specifically, as the incident wave height ranges from H

_{i}= 1.0 m to H

_{i}= 6.0 m, the maximum total mooring force exhibits an increase from 1.822 × 10

^{7}N to 1.881 × 10

^{7}N, thereby highlighting the substantial influence of the incident wave height on the maximum total mooring force.

#### 5.1.4. Effect of Incident Wave Period

_{i}= 4.6 m, and d = 50 m. The correlation between the maximum total mooring force and the incident wave period is determined through numerical simulations, as depicted in Figure 11d. Generally, there is a positive correlation between the maximum total mooring force and the incident wave period. Specifically, as the incident wave period increases from T = 6.5 s to T = 9.0 s, the maximum total mooring force ranges from 1.845 × 10

^{7}N to 1.982 × 10

^{7}N, suggesting that the incident wave period is a significant determinant of the maximum total mooring force.

#### 5.1.5. Effect of Water Depth

_{i}= 4.6 m, T = 7.6 s. The correlation between the maximum total mooring force and the water depth is obtained based on numerical simulations, as shown in Figure 11e.

^{7}N to 1.945 × 10

^{7}N. This observation suggests that water depth is a significant factor influencing the maximum mooring force. Specifically, the maximum mooring force exhibits an upward trend with increasing water depth when d ≤ 60 m. However, at larger water depths, the maximum mooring force remains relatively stable, indicating that the impact of the water depth on the maximum total mooring force becomes negligible when the water depth reaches a certain threshold. Due to the increase in the water depth, the angle between the mooring chain and the horizontal external force has a large value. Thus, there is a need for a large mooring chain force to balance the external force.

#### 5.2. Empirical Formula

_{i}), incident wave period (T), and water depth (d).

_{max}), the ratio of the wind speed to the one hundred years return value (V/V

_{100}), the ratio of the current velocity to the velocity of the foundation current (u/u

_{0}), the ratio of incident wave height to the large cylinder diameter (H/D

_{1}), the ratio of the incident wave period to the natural period of the floating breakwater structure (T/T

_{0}), and the ratio of water depth to the diameter of the large cylinder (d/D

_{1}). The relationship between the dimensionless total mooring force and the influential parameters is expressed as follows:

_{max}= 400,000,000 N, V

_{100}= 35 m/s, u = 1.0 m/s, D

_{1}= 12.0 m, and T

_{0}= 9.6 s. Assuming the application of a power law, the relationship stated in Equation (10) can be further expressed as:

_{100}≤ 1.5, 0.5 ≤ u/u

_{0}≤ 2.5, 0.083 ≤ H/D

_{1}≤ 0.5, 0.67 ≤ T/T

_{0}≤ 0.94, 2.5 ≤ d/D

_{1}≤ 5.83. The correlation analysis of the predicted and numerical results is conducted, and the correlation coefficient is R = 0.913, with F

_{E}and F

_{N}denoting the predicted value and numerical results of the maximum total mooring force. The proposed empirical formula can be used to predict the total mooring force of the floating breakwater in practical engineering. See Figure 12.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparisons of transmission coefficient related to the wave period between numerical and experimental results: (

**a**) d = 0.20 m; (

**b**) d = 0.25 m; Xu et al. (2017) [26].

**Figure 2.**Model sketch of multi-cylinder floating breakwater: (

**a**) cross-section of the breakwater; (

**b**) 3D numerical model.

**Figure 3.**Comparisons of transmission coefficients related to the wave period between experimental and numerical results: (

**a**) H

_{i}= 0.09 m; (

**b**) H

_{i}= 0.15 m; Qiu (2017) [17].

**Figure 5.**Wave fields around the floating breakwaters for H

_{i}= 4.0 m and T = 6.0 s: (

**a**) rectangular box; (

**b**) single circular cylinder; and (

**c**) multi-buoys.

**Figure 6.**Comparisons of wave transmission coefficients between different types of floating breakwaters.

**Figure 7.**Wave fields around the floating breakwaters: (

**a**) D

_{1}= 6.0 m; (

**b**) D

_{1}= 7.0 m; (

**c**) D

_{1}= 8.0 m; (

**d**) D

_{1}= 9.0 m; (

**e**) D

_{1}= 10.0 m; (

**f**) D

_{1}= 12.0 m.

**Figure 8.**Wave elevation along the centerline of the floating breakwaters for H

_{i}= 2.0 m, and T = 6.0 s: (

**a**) D

_{1}= 6.0 m; (

**b**) D

_{1}= 7.0 m; (

**c**) D

_{1}= 8.0 m; (

**d**) D

_{1}= 9.0 m; (

**e**) D

_{1}= 10.0 m; (

**f**) D

_{1}= 12.0 m.

**Figure 9.**Effects of influential factors on the transmission coefficient of the floating breakwater: (

**a**) large cylinder diameter; (

**b**) small cylinder diameter; (

**c**) position angle of the small cylinder; (

**d**) incident wave period; and (

**e**) incident wave height.

**Figure 10.**Correlation analysis between the empirical and numerical results of the wave transmission coefficient (5.0 m ≤ D

_{1}≤ 12.0 m; 1.0 m ≤ D

_{2}≤ 2.0 m; 30° ≤ α ≤ 90°; 1.5m ≤ H ≤ 3.5 m; 4.0 s ≤ T ≤ 8.0 s).

**Figure 11.**Effects of main factors on the mooring chain forces: (

**a**) wind speed, (

**b**) current velocity, (

**c**) incident wave period, (

**d**) incident wave height, and (

**e**) water depth.

**Figure 12.**Correlation analysis between empirical and numerical results of the maximum total mooring force.

Floating Breakwater Type | Reference | d/L | H/L | B/L | K_{t} |
---|---|---|---|---|---|

Two pontoon types | Ikeno et al. (1988) [27] | 0.33 | 0.02 | 0.19 | 0.68 |

Single-box type | Ikeno et al. (1988) [27] | 0.33 | 0.02 | 0.19 | 0.88 |

Y-frame without pipe | Mani (2014) [30] | 0.16 | 0.01 | 0.17 | 0.92 |

Y-frame with pipe | Mani (2014) [30] | 0.46 | 0.10 | 0.17 | 0.46 |

Cage type | Murali & Mani (1997) [28] | 0.46 | 0.10 | 0.19 | 0.50 |

Box type with steel truss | Uzaki et al. (2011) [29] | 0.14 | 0.03 | 0.21 | 0.91 |

Porous breakwater | Ji et al. (2016) [21] | 0.38 | 0.06 | 0.19 | 0.86 |

Mesh cage breakwater | Ji et al. (2016) [21] | 0.38 | 0.06 | 0.19 | 0.88 |

Single-box type | Present study | 0.89 | 0.06 | 0.16 | 0.86 |

Multi-buoy type | Present study | 0.89 | 0.06 | 0.17 | 0.66 |

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## Share and Cite

**MDPI and ACS Style**

Zang, Z.; Fang, Z.; Qiao, K.; Zhao, L.; Zhou, T.
Numerical Study on Wave Dissipation and Mooring Force of a Horizontal Multi-Cylinder Floating Breakwater. *J. Mar. Sci. Eng.* **2024**, *12*, 449.
https://doi.org/10.3390/jmse12030449

**AMA Style**

Zang Z, Fang Z, Qiao K, Zhao L, Zhou T.
Numerical Study on Wave Dissipation and Mooring Force of a Horizontal Multi-Cylinder Floating Breakwater. *Journal of Marine Science and Engineering*. 2024; 12(3):449.
https://doi.org/10.3390/jmse12030449

**Chicago/Turabian Style**

Zang, Zhipeng, Zhuo Fang, Kuan Qiao, Limeng Zhao, and Tongming Zhou.
2024. "Numerical Study on Wave Dissipation and Mooring Force of a Horizontal Multi-Cylinder Floating Breakwater" *Journal of Marine Science and Engineering* 12, no. 3: 449.
https://doi.org/10.3390/jmse12030449