# Energy Efficiency Analysis of a Deformable Wave Energy Converter Using Fully Coupled Dynamic Simulations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Computational Modelling

#### 2.1.1. Computational Domain and Boundary Conditions

**u**or γ. γ represents the change in spatial variation of any fluid property, such as dynamic molecular viscosity, density and so on.

#### 2.1.2. Computational Method

#### 2.1.3. Fluid Solution

**v**is the velocity vector, P is pressure, ρ is the density, and

**τ**is the viscous stress in which μ is the dynamic viscosity, and g is the gravitational acceleration.

**τ**can be described by the following equation:

#### 2.1.4. Structural Solution

_{solid}is the density of solid material,

**u**is the displacement vector,

**F = I + (∇u)T**is the deformation gradient tensor,

**I**is the second-order identity tensor, and

**Σ**is the second Piola–Kirchhoff stress tensor, which is related to the Cauchy stress tensor σ by the following expression:

**v**.

#### 2.1.5. Fluid-Structure Interaction

**v**is the velocity vector,

**u**is the displacement vector,

**n**is the normal vector at the interface, and

**σ**is the stress tensor.

#### 2.1.6. Data Acquisition and Processing

_{0}of 0.33, 1 and 2. When f/f

_{0}is 1, the amplitude obtained by FFT is 0.0196 m, which is very close to the value of subtracting two peaks in the time domain, which also shows the reliability of this method.

_{FFT}obtained at each probe point. In addition, f/f

_{0}of 0.33 should not be involved in the calculation. This is an unstable vibration. When the calculation time is extended, the vibration at this frequency will disappear.

#### 2.2. Verifications

#### 2.2.1. Fluid and Structural Parameters

^{2}and d/gT

^{2}are 1.69 × 10

^{−3}and 4.25 × 10

^{−2}, respectively. The parameters of solids are similar to those of rubber because most of the current WECs based on flexible structures are made of rubber. The model-scale wave condition was chosen based on previous validation against experiments, including (a) wave transmission and reflection; (b) structural deformations in waves, which was reported in detail in [26]. However, the results will also be applicable on a larger scale following Froude scaling. Although the scaling will involve a mismatch in Reynolds numbers, the influence is expected to be minimal as inertial force is much more significant than viscous force in this application.

#### 2.2.2. Laminar Flow

^{5}, it is considered that there is no turbulent component in the fluid. In other words, the laminar flow model can be used in this model to reduce the computational cost. It has become common sense that the critical Reynolds number for laminar flow to become turbulent is between 3.2 × 10

^{5}and 3 × 10

^{6}. The equation to calculate Reynolds number is as follows:

^{5}), it is also difficult to reach the Reynolds number of 3.2 × 10

^{5}. Therefore, it can be considered that the laminar flow model is consistent with the actual situation. In addition, the Keulegan–Carpenter number is around 0.2, which is smaller than 1, indicating that the flow does not separate from the plate [36]; thus, no turbulence effect is expected in the present study. However, we note that this is an ideal case and turbulence can be ubiquitous in real oceans.

#### 2.2.3. Mesh Convergence Study for Wave Modelling

#### 2.2.4. Mesh Convergence Study for Structural Modelling

## 3. Results and Discussion

#### 3.1. Effect of Deployment Depth and Length of Flexible Structure on Energy Conversion Efficiency

#### 3.2. Relationship between AR and Energy Conversion Efficiency of Flexible Structure

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the case: a flat flexible structure arranged under water is subjected to an incident wave and produces elastic deformation. L and d are the total length and deployment depth of the device (All units of length are meters).

**Figure 3.**Solution algorithm for strong two-way coupling (reset the figure in [35]).

**Figure 5.**Example of vertical vibration displacement at a point of the flexible structure (The data in the red box is used for analysis).

**Figure 7.**Liquid level data at CPH 6 and 12 (compared with data obtained through Stokes second-order wave theory).

**Figure 8.**Combined fluid and solid mesh (red part for solid subdomain and blue part for fluid subdomain).

**Figure 9.**Periodic interaction between the flexible WEC and incoming waves (C063). The applied wave condition in this figure is T = 1.1 s and H = 0.03 m. In the subfigures, the water phase is beneath the free surface in red, while the air phase is above the free surface in blue.

**Figure 12.**Upward limit (solid line) and downward limit (dash line) of the flexible structure’s deformations with different ARs.

Wave Type | Wave Period [s] | Height [m] | Wave Depth [m] | Wave Length [m] |
---|---|---|---|---|

Stokes Second | 1.1 | 0.03 [m] | 1.5 | 1.889 |

Transport Model | Simulation Type | Density [kg/m^{3}] | Kinematic Viscosity [m^{2}/s] |
---|---|---|---|

Newtonian | Laminar | 1000 | 1 × 10^{−6} |

Flexible Material Type | Density [kg/m^{3}] | Young’s Modulus [Pa] | Poisson’s Ratio [-] |
---|---|---|---|

St. Venant–Kirchhoff Elastic | 1000 | 5 × 10^{7} | 0.3 |

CPH | 3 | 6 | 9 | 12 |
---|---|---|---|---|

RMSE | 7.12 × 10^{−7} | 7.26 × 10^{−7} | 6.97 × 10^{−7} | 7.14 × 10^{−7} |

Number of Cells | 114 | 190 | 380 | 608 |
---|---|---|---|---|

Energy conversion efficiency | 137.93% | 138.49% | 137.77% | 138.52% |

L/λ = 0.2 | L/λ = 0.25 | L/λ = 0.3 | L/λ = 0.4 | L/λ = 0.5 | L/λ = 0.6 | L/λ = 0.65 | L/λ = 0.7 | L/λ = 0.75 | L/λ = 0.8 | |
---|---|---|---|---|---|---|---|---|---|---|

d/H = 01 | C020 | C020.5 | C021 | C022 | C023 | C024 | C024.5 | C025 | C025.5 | C026 |

L = 0.3778 d = 0.03 | L = 0.4723 d = 0.03 | L = 0.5667 d = 0.03 | L = 0.7556 d = 0.03 | L = 0.9445 d = 0.03 | L = 1.1334 d = 0.03 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.03 | L = 1.4168 d = 0.03 | L = 1.5112 d = 0.03 | |

d/H = 02 | C030 | C030.5 | C031 | C032 | C033 | C034 | C034.5 | C035 | C035.5 | C036 |

L = 0.3778 d = 0.06 | L = 0.4723 d = 0.06 | L = 0.5667 d = 0.06 | L = 0.7556 d = 0.06 | L = 0.9445 d = 0.06 | L = 1.1334 d = 0.06 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.06 | L = 1.4168 d = 0.06 | L = 1.5112 d = 0.06 | |

d/H = 03 | C040 | C040.5 | C041 | C042 | C043 | C044 | C044.5 | C045 | C045.5 | C046 |

L = 0.3778 d = 0.09 | L = 0.4723 d = 0.09 | L = 0.5667 d = 0.09 | L = 0.7556 d = 0.09 | L = 0.9445 d = 0.09 | L = 1.1334 d = 0.09 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.09 | L = 1.4168 d = 0.09 | L = 1.5112 d = 0.09 | |

d/H = 04 | C050 | C050.5 | C051 | C052 | C053 | C054 | C054.5 | C055 | C055.5 | C056 |

L = 0.3778 d = 0.12 | L = 0.4723 d = 0.12 | L = 0.5667 d = 0.12 | L = 0.7556 d = 0.12 | L = 0.9445 d = 0.12 | L = 1.1334 d = 0.12 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.12 | L = 1.4168 d = 0.12 | L = 1.5112 d = 0.12 | |

d/H = 05 | C060 | C060.5 | C061 | C062 | C063 | C064 | C064.5 | C065 | C065.5 | C066 |

L = 0.3778 d = 0.15 | L = 0.4723 d = 0.15 | L = 0.5667 d = 0.15 | L = 0.7556 d = 0.15 | L = 0.9445 d = 0.15 | L = 1.1334 d = 0.15 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.15 | L = 1.4168 d = 0.15 | L = 1.5112 d = 0.15 | |

d/H = 06 | C070 | C070.5 | C071 | C072 | C073 | C074 | C074.5 | C075 | C075.5 | C076 |

L = 0.3778 d = 0.18 | L = 0.4723 d = 0.18 | L = 0.5667 d = 0.18 | L = 0.7556 d = 0.18 | L = 0.9445 d = 0.18 | L = 1.1334 d = 0.18 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.18 | L = 1.4168 d = 0.18 | L = 1.5112 d = 0.18 | |

d/H = 07 | C080 | C080.5 | C081 | C082 | C083 | C084 | C084.5 | C085 | C085.5 | C086 |

L = 0.3778 d = 0.21 | L = 0.4723 d = 0.21 | L = 0.5667 d = 0.21 | L = 0.7556 d = 0.21 | L = 0.9445 d = 0.21 | L = 1.1334 d = 0.21 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.21 | L = 1.4168 d = 0.21 | L = 1.5112 d = 0.21 | |

d/H = 08 | C090 | C090.5 | C091 | C092 | C093 | C094 | C094.5 | C095 | C095.5 | C096 |

L = 0.3778 d = 0.24 | L = 0.4723 d = 0.24 | L = 0.5667 d = 0.24 | L = 0.7556 d = 0.24 | L = 0.9445 d = 0.24 | L = 1.1334 d = 0.24 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.24 | L = 1.4168 d = 0.24 | L = 1.5112 d = 0.24 | |

d/H = 09 | C100 | C100.5 | C101 | C102 | C103 | C104 | C104.5 | C105 | C105.5 | C106 |

L = 0.3778 d = 0.27 | L = 0.4723 d = 0.27 | L = 0.5667 d = 0.27 | L = 0.7556 d = 0.27 | L = 0.9445 d = 0.27 | L = 1.1334 d = 0.27 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.27 | L = 1.4168 d = 0.27 | L = 1.5112 d = 0.27 | |

d/H = 10 | C110 | C110.5 | C111 | C112 | C113 | C114 | C114.5 | C115 | C115.5 | C116 |

L = 0.3778 d = 0.30 | L = 0.4723 d = 0.30 | L = 0.5667 d = 0.30 | L = 0.7556 d = 0.30 | L = 0.9445 d = 0.30 | L = 1.1334 d = 0.30 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.30 | L = 1.4168 d = 0.30 | L = 1.5112 d = 0.30 | |

d/H = 11 | C120 | C120.5 | C121 | C122 | C123 | C124 | C124.5 | C125 | C125.5 | C126 |

L = 0.3778 d = 0.33 | L = 0.4723 d = 0.33 | L = 0.5667 d = 0.33 | L = 0.7556 d = 0.33 | L = 0.9445 d = 0.33 | L = 1.1334 d = 0.33 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.33 | L = 1.4168 d = 0.33 | L = 1.5112 d = 0.33 | |

d/H = 12 | C130 | C130.5 | C131 | C132 | C133 | C134 | C134.5 | C135 | C135.5 | C136 |

L = 0.3778 d = 0.36 | L = 0.4723 d = 0.36 | L = 0.5667 d = 0.36 | L = 0.7556 d = 0.36 | L = 0.9445 d = 0.36 | L = 1.1334 d = 0.36 | L = 1.228 d = 0.03 | L = 1.3223 d = 0.36 | L = 1.4168 d = 0.36 | L = 1.5112 d = 0.36 | |

d/H = 13 | C140 | C140.5 | C141 | C142 | C143 | C144 | C144.5 | C145 | C145.5 | C146 |

L = 0.3778 d = 0.39 | L = 0.4723 d = 0.39 | L = 0.5667 d = 0.39 | L = 0.7556 d = 0.39 | L = 0.9445 d = 0.39 | L = 1.1334 d = 0.39 | L = 1.228 d = 0.09 | L = 1.3223 d = 0.39 | L = 1.4168 d = 0.39 | L = 1.5112 d = 0.39 |

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**MDPI and ACS Style**

Luo, C.; Huang, L.
Energy Efficiency Analysis of a Deformable Wave Energy Converter Using Fully Coupled Dynamic Simulations. *Oceans* **2024**, *5*, 227-243.
https://doi.org/10.3390/oceans5020014

**AMA Style**

Luo C, Huang L.
Energy Efficiency Analysis of a Deformable Wave Energy Converter Using Fully Coupled Dynamic Simulations. *Oceans*. 2024; 5(2):227-243.
https://doi.org/10.3390/oceans5020014

**Chicago/Turabian Style**

Luo, Chen, and Luofeng Huang.
2024. "Energy Efficiency Analysis of a Deformable Wave Energy Converter Using Fully Coupled Dynamic Simulations" *Oceans* 5, no. 2: 227-243.
https://doi.org/10.3390/oceans5020014