# Numerical Study on the Mooring Force in an Offshore Fish Cage Array

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

_{n}is the solidity of the net, d is the thickness of one twine, and L is the length of the mesh bar. Due to the lack of data on the wake effect of currents and waves, the current velocity reduction after each cage is assumed as 15% from the experiment under uniform flow [5].

_{1}is wave height, h is the depth of the fluid, ω is the angular frequency of the wave, x and z are coordinates of the point.

_{M}is the mass coefficient, C

_{d}is drag coefficient, $\rho $ is the density of seawater, D is the diameter of the slender structure, u and $\dot{u}$ are the velocity and acceleration of the water particle, $\dot{x}$ and $\ddot{x}$ are the velocity and acceleration of the structure element.

_{a}is assumed to be 0. The pretension is set as 1 kN. The location of the mooring line (x, z) as a function of s can be expressed as [39]:

_{b}and f

_{H}are the load including gravity, buoyancy and hydrodynamic load from currents and waves, respectively.

## 3. Results

**Figure 4.**The instantaneous axial force distribution, unit: N. (current velocity = 0.5 m/s, wave height = 1 m, wave period = 10 s, time = 120 s).

#### 3.1. Numerical Model Subjected to Different Current Velocities

#### 3.2. Numerical Model Subjected to Waves of Different Heights

#### 3.3. Numerical Model Subjected to Different Attack Angles

## 4. Discussion

#### 4.1. Mooring Force Distribution

#### 4.2. Approximation of Wake Flow in APDL

_{0}. The drag force in the Morison equation can be expressed as:

## 5. Conclusions

- The cages deform more and move further away from their initial position with the increase of the current velocity.
- Both the mean values and the amplitudes of mooring forces increase with the current velocity and wave height. The tops of the upstream main ropes have the maximum mooring force under all sea states, which is also the connection location of multi-components. It should be the most concerned location for future research.
- The mooring forces on the frame ropes are the smallest while most frame ropes are slack. The downstream frame ropes do not play a structural role and can be removed.
- In most cases, the two adjacent bridle ropes do not pull the cage together, as the mooring force on one bridle rope is usually much larger than the other one. Thus, the two adjacent ropes do not always effectively reduce the stress concentration.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Deformation of fish cages under waves with a period of 10 s and a height of 1 m and different current velocities: (

**a**) current velocity = 0.25 m/s, (

**b**) current velocity = 0.5 m/s, (

**c**) current velocity = 0.75 m/s, (

**d**) current velocity = 1 m/s.

**Figure 7.**Deformation of upstream mooring cables: (

**a**) under current velocity of 0.25 m/s, (

**b**) under current velocity of 1 m/s).

**Figure 8.**Locations of the maximum mooring force (black: on main ropes, red: on bridle ropes, green: on frame ropes).

Parameters | Original Structure | Adjusted Structure |
---|---|---|

Density (kg/m^{3}) | 1140 | 1140 |

Young’s modulus (GPa) | 1.4 | 1.4 |

Twine diameter (mm) | 2.7 | N.A. |

Length of the mesh bar (mm) | 24 | N.A. |

Total Number of horizontal lines | 417 | 9 |

Total Number of vertical lines | 3140 | 20 |

Cross-section of each mesh bar (mm^{2}) | 5.73 | 265.07 (Horizontal lines) 899.37 (Vertical lines) |

Component | Dimension/Parameter | Value |
---|---|---|

Collar | Material type | HDPE |

Young’s modulus (GPa) | 0.65 | |

Diameter (m) | 24 | |

Cross-section diameter (m) | 0.5 | |

Thickness (mm) | 50 | |

Net | Material type | Nylon |

Young’s modulus (GPa) | 1.4 | |

Twine thickness (mm) | 2 | |

Solidity | 0.21 | |

Density (kg/m^{3}) | 1140 | |

Bottom ring | Material type | HDPE |

Young’s modulus (GPa) | 0.65 | |

Weight/length (kg/m) | 32 | |

Buoy | Buoyancy (kN) | 3 |

Bridle/frame rope | Material type | HMPE |

Stiffness, AE (N) | 1.25 × 10^{8} | |

Frame rope depth (m) | 3 | |

Distance between cages (m) | 6 | |

Submerged weight/length (N/m) | 9.92 | |

Breaking strength (kN) | 920 | |

Main rope | Material type | Six-strand wire rope (IWRC) HMPE |

Stiffness, AE (N) | 7.2 × 10^{7} | |

Anchor depth (m) | 50 | |

Submerged weight/length (N/m) | 54.4 | |

Breaking strength (kN) | 840 |

**Table 3.**The mean values and amplitudes of the maximum mooring forces in different parts of the mooring system under different current velocities.

Current Velocity (m/s) | 0.25 | 0.5 | 0.75 | 1 |
---|---|---|---|---|

Mean value_Main ropes (kN) | 21.88 | 48.41 | 80.97 | 116.52 |

Amplitude_Main ropes (kN) | 5.08 | 7.26 | 11.39 | 14.36 |

Mean value_Frame ropes (kN) | 4.50 | 12.68 | 22.86 | 35.23 |

Amplitude_Frame ropes (kN) | 1.83 | 1.20 | 1.02 | 2.42 |

Mean value_Bridle ropes kN) | 11.44 | 22.30 | 35.72 | 49.10 |

Amplitude_Bridle ropes (kN) | 2.13 | 4.26 | 6.69 | 6.73 |

Current Velocity (m/s) | 0.5 | 0.75 | 1 |
---|---|---|---|

Mean value_Bridle rope 1 (kN) | 22.29 | 35.73 | 49.10 |

Mean value_Bridle rope 2 (kN) | 0.15 | 0.18 | 0.28 |

Mean value_Bridle rope 3 (kN) | 0.18 | 0.26 | 0.50 |

Mean value_Bridle rope 4 (kN) | 17.89 | 29.55 | 42.02 |

**Table 5.**Mean values and amplitudes of the maximum mooring forces in different parts of the mooring system under different wave heights.

Wave Height (m/s) | 0.5 | 0.75 | 1 |
---|---|---|---|

Mean value_Main ropes (kN) | 44.73 | 45.97 | 48.41 |

Amplitude_Main ropes (kN) | 3.40 | 5.10 | 7.26 |

Mean value_Frame ropes (kN) | 11.91 | 12.12 | 12.68 |

Amplitude_Frame ropes (kN) | 0.83 | 0.95 | 1.2 |

Mean value_Bridle ropes kN) | 20.49 | 21.09 | 22.30 |

Amplitude_Bridle ropes (kN) | 2.11 | 3.07 | 4.26 |

**Table 6.**Mean values and amplitudes of the mooring forces on main ropes under different attack angles.

Attack Angle (°) | 0 | 22.5 | 45 | 67.5 | 90 |
---|---|---|---|---|---|

Mean value_Main rope 1 (kN) | 48.41 | 44.65 | 29.74 | 12.20 | 4.49 |

Amplitude_Main rope 1 (kN) | 7.26 | 8.19 | 11.6 | 3.12 | 0.38 |

Mean value_Main cable 2 (kN) | 22.09 | 29.41 | 22.83 | 8.43 | 3.63 |

Amplitude_Main rope 2 (kN) | 4.18 | 7.03 | 5.90 | 1.96 | 0.31 |

Mean value_Main cable 3 (kN) | 3.55 | 7.36 | 16.38 | 25.52 | 18.92 |

Amplitude_Main cable 3 (kN) | 0.31 | 1.86 | 7.57 | 8.5 | 5.43 |

Mean value_Main cable 4 (kN) | 5.95 | 24.46 | 45.56 | 42.32 | 38.49 |

Amplitude_Main cable 4 (kN) | 0.71 | 7.08 | 14.33 | 12.2 | 10.86 |

Location of max force | 1 | 1 | 4 | 4 | 4 |

**Table 7.**Mean values and amplitudes of the mooring forces on bridle ropes under different attack angles.

Attack Angle (°) | 0 | 22.5 | 45 | 67.5 | 90 |
---|---|---|---|---|---|

Mean value_Bridle rope 5 (kN) | - | 2.06 | 11.07 | 22.22 | 19.18 |

Amplitude_Bridle rope 5 (kN) | - | 1.76 | 6.70 | 7.39 | 5.42 |

Mean value_Bridle rope 6 (kN) | 22.29 | 31.59 | 25.25 | 9.82 | - |

Amplitude_Bridle rope 6 (kN) | 4.26 | 7.19 | 6.54 | 3.31 | - |

Mean value_Bridle rope 7 (kN) | 9.62 | 18.31 | 15.79 | 1.50 | - |

Amplitude_Bridle rope 7 (kN) | 0.72 | 3.82 | 2.99 | 1.29 | - |

Mean value_Bridle rope 8 (kN) | - | - | 8.09 | 16.21 | 15.47 |

Amplitude_Bridle rope 8 (kN) | - | - | 7.82 | 5.05 | 5.30 |

Mean value_Bridle rope 9 (kN) | - | 5.65 | 11.28 | 13.44 | 18.62 |

Amplitude_Bridle rope 9 (kN) | - | 2.59 | 2.66 | 5.07 | 6.18 |

Mean value_Bridle rope 10 (kN) | 6.00 | 15.98 | 25.06 | 12.05 | - |

Amplitude_Bridle rope 10 (kN) | 1.40 | 7.06 | 7.01 | 3.52 | - |

Location of max force | 6 | 6 | 6 | 5 | 5 |

**Table 8.**Mean values and amplitudes of the mooring forces on frame ropes under different attack angles.

Attack Angle (°) | 0 | 22.5 | 45 | 67.5 | 90 |
---|---|---|---|---|---|

Mean value_Frame rope 11(kN) | 12.68 | 9.32 | 2.77 | - | - |

Amplitude_Frame rope 11(kN) | 1.2 | 1.58 | 0.72 | - | - |

Mean value_Frame rope 12 (kN) | - | - | 1.97 | 4.16 | 3.38 |

Amplitude_Frame rope 12 (kN) | - | - | 1.03 | 1.35 | 0.94 |

Mean value_Frame rope 13 (kN) | 12.31 | 10.48 | - | - | - |

Amplitude_Frame rope 13 (kN) | 1.88 | 1.36 | - | - | - |

Mean value_Frame rope 14 (kN) | - | 4.29 | - | - | - |

Amplitude_Frame rope 14 (kN) | - | 3.23 | - | - | - |

Location of max force | 11 | 13 | 11 | 12 | 12 |

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**MDPI and ACS Style**

Liu, Z.; Wang, S.; Guedes Soares, C. Numerical Study on the Mooring Force in an Offshore Fish Cage Array. *J. Mar. Sci. Eng.* **2022**, *10*, 331.
https://doi.org/10.3390/jmse10030331

**AMA Style**

Liu Z, Wang S, Guedes Soares C. Numerical Study on the Mooring Force in an Offshore Fish Cage Array. *Journal of Marine Science and Engineering*. 2022; 10(3):331.
https://doi.org/10.3390/jmse10030331

**Chicago/Turabian Style**

Liu, Zhongchi, Shan Wang, and C. Guedes Soares. 2022. "Numerical Study on the Mooring Force in an Offshore Fish Cage Array" *Journal of Marine Science and Engineering* 10, no. 3: 331.
https://doi.org/10.3390/jmse10030331