# Peak Forces on Wave Energy Linear Generators in Tsunami and Extreme Waves

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Governing Equations

#### 2.2. Numerical Implementation

#### 2.2.1. Numerical Wavetank

#### 2.2.2. Incident Regular Wave

#### 2.2.3. Incident Tsunami Wave

#### 2.3. The WEC Model

## 3. Results

#### 3.1. Peak Forces as a Function of Wave Height

#### 3.2. The Impact of Friction on the Survivability in Extreme Waves

#### 3.3. Endstop Forces Decrease with Increased Damping for Periodic Waves

#### 3.4. Influence of Line Length on Endstop Peak Force

## 4. Discussion

## 5. Conclusions

- For periodic waves, it was seen that both increased linear damping, ${F}_{PTO}=\gamma \dot{r}$, and increased constant damping ${F}_{fric}=constant$, decreased the force of the endstop hits. This corresponds well with established experimental results.
- If the incident wave was not periodic, for the tsunami event or the transient waves at the front of the regular wave train, it was seen that increased friction could result in a latching effect and actually increase the force of the endstop hit instead of decreasing it. It is possible that this effect could also occur for irregular waves during normal operating conditions.
- Due to the differences in fluid velocity fields, the WEC was more vulnerable to a too long line length when impacted by a tsunami wave than by a regular wave. For a regular wave, an increased line length resulted in lower endstop forces and decreased surge motion. For the tsunami wave on the other hand, an increased line length resulted in significantly higher endstop forces.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

WEC | Wave Energy Converter |

CFD | Computational Fluid Dynamics |

RANS | Reynolds Average Navier-Stokes |

VOF | Volume of Fluid |

PTO | Power Take Off |

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**Figure 2.**The mesh is refined in a box surrounding the surface, and further refined in the vicinity of the buoy. The color represents the scalar field $\alpha $. Blue represents $\alpha =0$ (air), while red represents $\alpha =1$ (water).

**Figure 3.**For the dam-break approach, the simulation domain is elongated and a water volume is placed on top of the initially still water surface.

**Figure 4.**(

**a**) The WEC is modeled as a floating buoy, restrained by a force in the connection line. The force is directed along a vector pointing at the fixed anchoring position; (

**b**) The generator has a limited stroke length. It is also seen that the translator-stator overlap is not full during the whole stroke length.

**Figure 5.**Surface elevation of the modeled incident waves. The wave period was 8.5 s, and five different wave heights were simulated.

**Figure 6.**The peak forces plotted as a function of wave height. The peak forces decrease with increasing generator damping for damping factors up to $\gamma =100$ kNs/m. For higher $\gamma $, the generator is strong enough to keep the translator from hitting the upper endstop and the peak forces are not further reduced.

**Figure 7.**The total force in the connection line is a sum of of ${F}_{PTO}{A}_{frac}^{2}$, ${F}_{fric}$, ${F}_{spring}$, ${F}_{endstop}$ and the gravitational force. The line force is compared for three levels of generator damping $\gamma $. The corresponding translator position and speed are also seen. The surface elevation of the incident wave is seen together with the translator position. The horizontal dashed and solid lines mark the free and the total stroke length, respectively.

**Figure 8.**The force in the connection line when the WEC is impacted by a high regular wave. Five levels of friction are compared.

**Figure 9.**This is a zoom of Figure 8. The WEC is impacted by a high regular wave. The solid horizontal line marks the bottom of the generator, and the dashed horizontal line marks the lower endstop spring.

**Figure 10.**The force in the connection line and the translator position when the WEC is impacted by a tsunami event. Five levels of friction are compared.

**Figure 11.**The WEC impacted by the tsunami wave. The figure shows the four main events: The first peak force occurred at t = 8.5 s, one second before the first wave peak propagated over the WEC at t = 10.5 s. The second peak force occurred at t = 14.5 s, one second before the second wave peak at t = 15.5 s. The origin marks the initial position of the buoy.

**Figure 12.**The line force and the translator position when the WEC is impacted by a high periodic wave, comparing the influence of generator damping factor $\gamma $. Increasing the generator damping decreases the force of the endstop hits. For $\gamma =80$ kNs/m or higher, endstop hits were prevented.

**Figure 13.**The line force and translator position for different levels of frictional damping. This figure shows the third wave peak of Figure 8. The generator damping factor is constant at $\gamma =40$ kNs/m. The force of the endstop hit is reduced by an increased friction.

**Figure 14.**This figure is reprinted with permission from reference [24]. In a physical wave tank experiment, the line force of a WEC was studied when different levels of frictional damping were applied. The incident wave had a height of 5.7 m and an embedded focused extreme wave hit the WEC at t = 134 s. Three levels of friction was applied: 18 kN (

**left**), 59 kN (

**middle**), and 83 kN (

**right**). The force in the endstop hits decrease with increased damping.

**Figure 15.**A regular wave was modeled, comparing the influence of different line lengths. The generator damping factor was $\gamma =40$ kNs/m and the friction was set to ${F}_{fric}$ = 5 kN.

**Figure 16.**Line force, translator position and buoy position when the WEC is impacted by a tsunami wave. The influence of line length is compared.

**Figure 17.**The WEC impacted by a regular wave (

**left**), and by a tsunami wave (

**right**). The incident waves are traveling from the left to the right in the figures. The connection line length is modelled as too short in those simulations. The origin marks the original position of the buoy in still water.

Parameter | Abbrevation | Value |
---|---|---|

Buoy radius | ${R}_{buoy}$ | 1.7 m |

Buoy height | ${h}_{buoy}$ | 2.1 m |

Buoy mass | ${m}_{buoy}$ | 5700 kg |

Translator mass | ${m}_{translator}$ | 6500 kg |

Translator height | ${h}_{translator}$ | 3 m |

Stator height | ${h}_{stator}$ | 2 m |

Free stroke length up/down | ${l}_{freestroke}^{up/down}$ | 1 m |

Total stroke length up/down | ${l}_{totstroke}^{up/down}$ | 1.25 m |

Endstop spring constant | ${\kappa}_{endstop-spring}$ | 250 kN/m |

Spring constant corresponding to line elasticity | ${\kappa}_{line}$ | 2600 kN/m |

Wave Height | 3 m | 4 m | 5 m | 6 m | 7 m |

Wave Steepness (kA) | 0.08 | 0.11 | 0.14 | 0.17 | 0.19 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sjökvist, L.; Göteman, M.
Peak Forces on Wave Energy Linear Generators in Tsunami and Extreme Waves. *Energies* **2017**, *10*, 1323.
https://doi.org/10.3390/en10091323

**AMA Style**

Sjökvist L, Göteman M.
Peak Forces on Wave Energy Linear Generators in Tsunami and Extreme Waves. *Energies*. 2017; 10(9):1323.
https://doi.org/10.3390/en10091323

**Chicago/Turabian Style**

Sjökvist, Linnea, and Malin Göteman.
2017. "Peak Forces on Wave Energy Linear Generators in Tsunami and Extreme Waves" *Energies* 10, no. 9: 1323.
https://doi.org/10.3390/en10091323