# Numerical Investigation of the Scaling Effects for a Point Absorber

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## Abstract

**:**

## 1. Introduction

#### Wave Energy Converter

^{5}in order to neglect the viscous effects for all type of wave energy converters. Nevertheless, scale effects have been studied separately for different types of WECs with varying results. Schmitt and Elsaber [18] give a detailed discussion on the suitability of Froude scaling laws for a Oscillating Wave Surge Converter (OWSC) where they numerically test its behaviour by scaling the fluid viscosity. In this study they conclude that the differences in flow patterns for different scales can be explained by the changes in viscosity, nevertheless they state that some uncertainties remain related to the mesh influence. Palm et al. [19] analyse the nonlinear forces on a moored point absorber wave energy converter (PA-WEC) at prototype and model scale using CFD for two wave conditions, finding an amplitude response reduction between 1 and 4% due to viscous forces and between 18 and 19% due to induced drag, non-linear added mass, and radiation forces. This study was made for two regular 5th order waves in stationary state. Recently, Windt et al. [20,21] studied firstly the scale effect of a moored PA-WEC device exposed to focused waves, finding differences around 5% between different scales, and secondly they studied the hydrodynamic scaling effects for the wavestar WEC exposed to regular and irregular stationary waves, showing that significant scaling effects occur for the viscous component of the hydrodynamic loads on the WEC hull, stating that additional studies are required for extreme events, e.g., breaking waves. In the other hand, in order to simulate trains of regular waves, most authors recommend the selection of a wave theory based on dimensional parameters, where two of the most cited works are from Le Méhauté in 1976 and Hedges in 1995 [22,23]. Based on these recommendations, several authors propose numerical models using, from linear theory [24,25,26,27] to higher order wave theories [28,29], to simulate the wave behaviour. In some of these studies the goal was to numerically model scaled WEC devices where the same wave theory selection map from Le Méhauté was used without considering the influence of the scale effect on the correct selection of the wave theory [26,27]. Hence, the main goal of the presented work is to study the scale effect related to a one DOF point absorber subjected to three different waves and to study how the wave theory applied to these waves influences the predicted WEC behaviour by using different scales. Section 2 provides the theoretical background, which is used for the numerical model presented in Section 3. The gained results are discussed in Section 4 followed by the conclusions in Section 5.

## 2. Wave Theories

#### 2.1. Airy Wave Theory

#### 2.2. Stokes Wave Theories

#### Validity Regions

## 3. Numerical Model

#### 3.1. Numerical Wave Tank Characteristics

#### 3.2. Solver Settings

#### 3.3. Boundary Conditions

#### 3.4. Mesh

- Case A: Waves ${w}_{2}$, ${w}_{3}$, and ${w}_{4}$ without WEC, for the scale 1:1 and 1:50;
- Case B: Waves ${w}_{2}$, ${w}_{3}$, and ${w}_{4}$, including the WEC, for the scale 1:1 and 1:50.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

DOF | Degree Of Freedom |

FSE | Free Surface Elevation |

LWT | Linear Wave Theory |

OWC | Oscillating Wave Surge Converter |

OCWBC | Open Channel Wave Boundary Condition |

PA-WEC | Point Absorber Wave Energy Converter |

RANS | Reynolds-Averaged Navier-Stokes |

SPH | Smoothed-particle hydrodynamics |

TLR | Technology readiness level |

VOF | Volume of Fluid |

WEC | Wave Energy Converter |

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**Figure 4.**Mesh convergence analysis using free surface elevation at the middle of the width ${W}_{t}$ and at one wavelength from the inlet, for Case A.

**Figure 5.**Mesh convergence analysis using free surface elevation at the middle of the width ${W}_{t}$ and at one wavelength from the inlet, for Case B.

**Figure 7.**Free surface elevation at the middle of the width ${W}_{t}$ and at one wavelength from the inlet for cases ${w}_{2}$, ${w}_{3}$, and ${w}_{4}$, using recommended order wave theory, for scales 1:1 and upscaled 1:50, using the wave tank without WEC at the left hand side and with WEC at the right hand side.

**Figure 8.**Heave displacement for cases ${w}_{2}$, ${w}_{3}$, and ${w}_{4}$, using recommended order wave theory, for scales 1:1 and upscaled 1:50.

**Figure 9.**Normalised difference $\epsilon $ between heave displacement of the WEC. (

**a**) Normalised difference between the same order model of scale 1:50 and scale 1:1. (

**b**) Normalised difference between each model in scale 1:50 and recommended scale 1:1. (

**c**) Normalised difference between different order models for scale 1:1. (

**d**) Normalised difference between different order models for scale 1:50.

**Figure 10.**Heave displacement for cases ${w}_{2}$, ${w}_{3}$ and ${w}_{4}$ using 1st order, 2nd order, 3rd order and 4th order wave theories, for scales 1:1 and 1:50.

**Table 1.**Dimensions of the tank illustrated in Figure 1 in meters for the different scales.

Scale | h | ${\mathit{h}}_{\mathit{m}}$ | ${\mathit{W}}_{\mathit{t}}$ | L | ${\mathit{L}}_{\mathit{d}}$ | ${\mathit{L}}_{\mathit{b}}$ | ${\mathit{D}}_{\mathit{b}}$ | ${\mathit{f}}_{\mathit{b}}$ |
---|---|---|---|---|---|---|---|---|

1:1 | 15 | 29.58 | 24 | 82.0 | 44.5 | 3 | 3.9 | 1.1695 |

1:50 | 0.300 | 0.5916 | 0.48 | 1.64 | 0.89 | 0.06 | 0.078 | 0.02339 |

**Table 2.**Characteristics of the studied point absorber at laboratory scale shown in Figure 2.

Parameter | Value | Units |
---|---|---|

Total mass | 0.235 | Kg |

Construction method | 3D printed | - |

Material | Polylactic Acid (PLA) | - |

Surface treatment | Epoxy adhesive | - |

Support method | Axial bearings | - |

Parameter | Value | Units |
---|---|---|

Time step | adaptive | s |

Turbulence model | realisable $k-\u03f5$ | - |

WEC density | 574 | kg/m^{3} |

Water-Air surface tension | 0.074 | mN/m |

Scale | Height $\left(\mathit{H}\right)$ | Length ($\mathit{\lambda}$) | Period ($\mathit{\tau}$) | $\mathit{h}/\mathit{g}{\mathit{\tau}}^{2}$ | $\mathit{H}/\mathit{g}{\mathit{\tau}}^{2}$ | |
---|---|---|---|---|---|---|

${w}_{4}$ | 1:1 | 4.200 m | 30 m | 4.394 s | 0.0793 | 0.0222 |

1:50 | 0.084 m | 0.6 m | 0.621 s | 0.0793 | 0.0222 | |

${w}_{3}$ | 1:1 | 2.400 m | 30 m | 4.394 s | 0.0793 | 0.0127 |

1:50 | 0.048 m | 0.6 m | 0.621 s | 0.0793 | 0.0127 | |

${w}_{2}$ | 1:1 | 1.000 m | 30 m | 4.394 s | 0.0793 | 0.0053 |

1:50 | 0.020 m | 0.6 m | 0.621 s | 0.0793 | 0.0053 |

Zone | Boundary Condition |
---|---|

Top | Pressure Outlet |

Inlet | Velocity Inlet and OCWBC |

Outlet | Pressure Outlet and OCWBC |

Walls and buoy | No-slip Wall |

**Table 6.**The mesh element number divided by the wavelength ($\lambda $) case A and B and the two scales, in total zone, refined zone 1 for the case A, and refined zone 1 + refined zone 2 for case B.

N° Ele/$\mathit{\lambda}$ | Scale | Mesh 1 | Mesh 2 | Mesh 3 | |||
---|---|---|---|---|---|---|---|

Total | Zone 1 | Total | Zone 1 | Total | Zone 1 | ||

Case A | 1:1 | 9758 | 3955 | 13,292 | 5707 | 3736 | 23,575 |

1:50 | 350,334 | 142,026 | 664,648 | 285,439 | 2,007,075 | 1,270,300 | |

Total | Zone 1 + 2 | Total | Zone 1 + 2 | Total | Zone 1 + 2 | ||

Case B | 1:1 | 14,265 | 10,613 | 17,001 | 14,445 | 34,560 | 27,661 |

1:50 | 715,819 | 532,584 | 1,284,762 | 1,025,800 | 1,434,264 | 1,148,041 |

Scale | Wave | Mesh 1/Mesh 2 | Mesh 2/Mesh 3 | |
---|---|---|---|---|

Case A | 1:1 | ${w}_{2}$ | 0.470% | 0.079% |

${w}_{3}$ | 0.036% | 0.109% | ||

${w}_{4}$ | 0.107% | 1.152% | ||

1:50 | ${w}_{2}$ | 0.016% | 0.202% | |

${w}_{3}$ | 0.014% | 0.021% | ||

${w}_{4}$ | 1.041% | 0.434% | ||

Case B | 1:1 | ${w}_{2}$ | 0.035% | 0.452% |

${w}_{3}$ | 0.213% | 0.085% | ||

${w}_{4}$ | 0.292% | 0.021% | ||

1:50 | ${w}_{2}$ | 2.073% | 0.322% | |

${w}_{3}$ | 0.234% | 0.846% | ||

${w}_{4}$ | 0.875% | 0.034% | ||

Heave | 1:1 | ${w}_{2}$ | 0.012% | 1.931% |

${w}_{3}$ | 0.134% | 0.705% | ||

${w}_{4}$ | 0.383% | 0.473% | ||

1:50 | ${w}_{2}$ | 2.076% | 0.854% | |

${w}_{3}$ | 0.348% | 0.034% | ||

${w}_{4}$ | 0.871% | 0.084% |

Parameter | Froude Scaling Ratio |
---|---|

Length | $\Lambda $ |

Time | ${\Lambda}^{1/2}$ |

Mass | ${\Lambda}^{3}$ |

Power | ${\Lambda}^{7/2}$ |

**Table 9.**Normalised difference $\epsilon $ between scales, for the three cases analysed using the recommended wave theory.

${\mathit{w}}_{2}$ | ${\mathit{w}}_{3}$ | ${\mathit{w}}_{4}$ | |
---|---|---|---|

Case A FSE | 1.09% | 0.58% | 3.05% |

Case B FSE | 6.46% | 3.00% | 0.59% |

Case B Heave | 38.0% | 30.0% | 56.0% |

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**MDPI and ACS Style**

Pierart, F.G.; Fernandez, J.; Olivos, J.; Gabl, R.; Davey, T.
Numerical Investigation of the Scaling Effects for a Point Absorber. *Water* **2022**, *14*, 2156.
https://doi.org/10.3390/w14142156

**AMA Style**

Pierart FG, Fernandez J, Olivos J, Gabl R, Davey T.
Numerical Investigation of the Scaling Effects for a Point Absorber. *Water*. 2022; 14(14):2156.
https://doi.org/10.3390/w14142156

**Chicago/Turabian Style**

Pierart, Fabián G., Joaquín Fernandez, Juan Olivos, Roman Gabl, and Thomas Davey.
2022. "Numerical Investigation of the Scaling Effects for a Point Absorber" *Water* 14, no. 14: 2156.
https://doi.org/10.3390/w14142156