# Far-Field Maximal Power Absorption of a Bulging Cylindrical Wave Energy Converter

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. Far-Field Maximal Absorption Width without Constraint

#### 2.2. Far-Field Maximal Absorption Width under Constraint

#### 2.3. Modal Analysis of the SBM Offshore’s S3 Device

## 3. Numerical Results

#### 3.1. Implementation and Setup

#### 3.2. Mesh Convergence Study

#### 3.3. Examples of Deformation Profiles

#### 3.4. Maximal Absorption Width with and without Constraint

#### 3.5. Comparison with the Linear Response of the WEC

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The relative error on the maximal absorption width computed with the first four modes and the first fifteen modes of the S3 with respect to a reference solution with 11850 panels, for several wavelengths and with and without constraint on the maximal deformation ($b\to \infty $ means no constraint). Table 1 gives the parameters of the problem.

**Figure 4.**Cross section at a given time of a cylinder with length $L=60$ m and radius ${r}_{S}=0.9$ m following its maximal motion predicted while using the first five modes of deformation for waves of wavelength $\lambda =0.8\phantom{\rule{3.33333pt}{0ex}}L$ traveling in the direction of the device, for several incoming wave amplitudes $|\widehat{A}|$ and several constraints b.

**Figure 5.**Dimensionless maximal absorption width as a function of the wavelength of the incoming waves. Each line corresponds to the absorption width of the different degrees of freedom of the S3 taken individually. The dotted lines are the unconstrained absorption widths. The solid lines are the absorption width under the constraint $b=1.0$.

**Figure 6.**Dimensionless maximal absorption width as a function of the wavelength of the incoming waves. Each line corresponds to the absorption width for the combination of several modes. The dotted lines are the unconstrained absorption widths. The solid lines are the absorption widths under the constraint $b=1.0$.

**Figure 7.**Dimensionless maximal absorption width as a function of the direction of the incoming waves. Each line corresponds to the absorption width for the combination of several modes. The dotted lines are the unconstrained absorption widths. The solid lines are the absorption widths under the constraint $b=1.0$.

**Figure 8.**A wave with wavelength $\lambda $ propagating in the direction of the cylinder is exciting the mode of wavelength $\lambda $. When the same wavelength is propagating with an angle $\beta $ with respect to the cylinder, it can excite a mode of wavelength $\lambda /cos\left(\beta \right)$.

**Figure 9.**Dimensionless absorption width as a function of the wavelength of the incoming waves for the optimal deformation under constraint and the linear response for two values of $\eta $. The solid lines represent the constrained maximal absorption widths. The dash-dotted lines correspond to the linear response.

length L | 60 $\mathbf{m}$ |
---|---|

radius ${r}_{S}$ | $0.9\text{}$$\mathrm{m}$ |

cross section ${S}_{s}$ | $2.544\text{}$${\mathrm{m}}^{2}$ |

${z}_{S}$ | $-1.35\text{}$$\mathrm{m}$ |

D | $2.248\times {10}^{-5}$ ${\mathrm{Pa}}^{-1}$ |

${T}_{s}$ | $3.8\times {10}^{4}$$\mathrm{N}$ |

M | 38,000 kg |

$\eta $ | $0.323$ ${\mathrm{m}}^{2}$${\mathrm{s}}^{-2}$ |

${B}_{R}$ | $8\pi \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-6}$ |

water density $\rho $ | 1025 $\mathrm{k}\mathrm{g}$/${\mathrm{m}}^{-3}$ |

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**MDPI and ACS Style**

Ancellin, M.; Dong, M.; Jean, P.; Dias, F. Far-Field Maximal Power Absorption of a Bulging Cylindrical Wave Energy Converter. *Energies* **2020**, *13*, 5499.
https://doi.org/10.3390/en13205499

**AMA Style**

Ancellin M, Dong M, Jean P, Dias F. Far-Field Maximal Power Absorption of a Bulging Cylindrical Wave Energy Converter. *Energies*. 2020; 13(20):5499.
https://doi.org/10.3390/en13205499

**Chicago/Turabian Style**

Ancellin, Matthieu, Marlène Dong, Philippe Jean, and Frédéric Dias. 2020. "Far-Field Maximal Power Absorption of a Bulging Cylindrical Wave Energy Converter" *Energies* 13, no. 20: 5499.
https://doi.org/10.3390/en13205499