# Smoothed Particle Hydrodynamics Simulation of a Mariculture Platform under Waves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. SPH Model

#### 2.1. SPH Method

^{3}, and ${c}_{0}$ is the speed of sound at the reference density.

#### 2.2. Boundary Conditions

#### 2.3. Wave Generation

#### 2.4. Wave Absorption

#### 2.5. Fluid-Driven Objects

**I**is the inertia moment,

**V**is the velocity,

**Ω**is the rotational velocity, and

**R**

_{0}is the mass center.

#### 2.6. Moorings

_{n}. The cable segment is numbered as S

_{i + 1/2}, which is between nodes i and i+1. Meanwhile, the forces on the node include the internal axial stiffness (

**T**), damping forces (

**C**), weight (

**G**), buoyancy forces (

**B**), hydrodynamic forces from Morison’s equation (

**D**), and vertical spring–damper forces from contact with the seabed, as shown in Figure 1b. The acceleration of each node can be calculated by solving the equation as shown below. Finally, we can derive the resultant force (

**F**) and torque (

_{m}**T**) of the mooring system acting on the floating box. The complete motion equation for node i is:

_{m}## 3. Validation and Simulation

#### 3.1. Decay Test: Theory vs. SPH

^{3}was half-immersed, and each side of the floating box was assumed to be 0.18 m. Figure 2 illustrates the numerical flume in the static equilibrium position. In this case, the initial particle spacing dp was 0.0025 m, leading to a smoothing length of 0.004243 m and 1,283,256 particles in 2-D.

^{2}); B is the dimension;${a}_{zz}$ is the added mass coefficient in the heave motion, which is equal to 0.9 according to experiments conducted on a half-immersed square floating box [39]; and M is the mass.

#### 3.2. Floating Body Test: Experiment vs. SPH

^{3}, resulting in a mass of 12.6 kg.

#### 3.3. Floating Body Test: Mooring vs. without Mooring

_{l}= 12 kg/m. A high cable stiffness, k

_{cable}= 4 × 10

^{5}N/m, was used to avoid stretching of the mooring lines. The time step was Δt

_{m}= 1 × 10

^{–4}s. To improve the computational efficiency, [1,20,41] developed codes for DualSPHysics with a CUDA toolkit using the GPU acceleration technique. The GPU card was an NVIDIA GeForce RTX 1070 Ti. The initial particle distance dp was 0.01, leading to a total particle number of 56,160. Simulating a 25 s physical process took 71,015 s.

## 4. Application

#### 4.1. 3-DoF under a Typical Annual Wave

#### 4.2. 3-DoF under Typhoon Dujuan Waves

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustration of the lumped-mass approach: (

**a**) mooring line discretization; (

**b**) internal and external forces on a cable.

**Figure 5.**Scope of wave theory. (Floating Body Test Wave #1 with H = 0.1 m T = 1.2 s; Typical annual wave simulation Wave#2 with H = 0.6m T = 5 s; Typhoon Dujuan simulation Wave#3 with H = 3.7 m T = 9.9 s; Typhoon experiments Wave#4 with H = 1.5 m, T = 7.4 s).

**Figure 6.**Different snapshots of the freely floating box under regular waves. Particle colors correspond to the horizontal flow velocity (

**left**) and pressure (

**right**) values.

**Figure 7.**Comparison between experimental and numerical time series of the 3-DoF motions under a regular wave with H = 0.1 m and T = 1.2 s.

**Figure 12.**Comparison between theoretical and numerical surface elevations under a regular wave with H = 0.6 m and T = 5 s.

**Figure 13.**Time series of 3-DoF motions of the mariculture platform under the regular and irregular waves with H = 0.6 m and T = 5 s simulated with SPH.

**Figure 14.**Time series of 3-DoF motions of the mariculture platform under the regular and irregular waves with H = 3.7 m, T = 9.9 s, and d = 11.88 m simulated with SPH.

**Figure 15.**Snapshot of the mariculture platform with H = 3.7 m, T = 9.9 s, and d = 11.88 m simulated with SPH. Particle colors correspond to the horizontal flow velocity.

**Figure 16.**Time series of 3-DoF motions of the mariculture platform under the regular wave with H = 1.5 m, T = 7.4 s, and d = 13.64 m, 10.78 m, and 7.93 m simulated with SPH.

Time step size | 1 × 10^{−4} s |

Segments | 20 |

Diameter | 0.01 m |

Mass per unit length | 12 kg/m |

Cable stiffness | 4 × 10^{5} N/m |

Points | Coordinates (x, z) |
---|---|

Anchor 1 | (8.5, −0.4) |

Fairlead 1 | (9.1, 0.0) |

Anchor 2 | (10, −0.4) |

Fairlead 2 | (9.4, 0.0) |

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**MDPI and ACS Style**

Zhang, F.; Zhang, L.; Xie, Y.; Wang, Z.; Shang, S.
Smoothed Particle Hydrodynamics Simulation of a Mariculture Platform under Waves. *Water* **2021**, *13*, 2847.
https://doi.org/10.3390/w13202847

**AMA Style**

Zhang F, Zhang L, Xie Y, Wang Z, Shang S.
Smoothed Particle Hydrodynamics Simulation of a Mariculture Platform under Waves. *Water*. 2021; 13(20):2847.
https://doi.org/10.3390/w13202847

**Chicago/Turabian Style**

Zhang, Feng, Li Zhang, Yanshuang Xie, Zhiyuan Wang, and Shaoping Shang.
2021. "Smoothed Particle Hydrodynamics Simulation of a Mariculture Platform under Waves" *Water* 13, no. 20: 2847.
https://doi.org/10.3390/w13202847