# Irregular Wave Validation of a Coupling Methodology for Numerical Modelling of Near and Far Field Effects of Wave Energy Converter Arrays

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generic Coupling Methodology

## 3. Application of the Coupling Methodology between the Wave Propagation Model, MILDwave, and the Wave–Structure Interaction Solver NEMOH for Irregular Waves

#### 3.1. The Wave Propagation Model, MILDwave and the Wave–Structure Interaction Solver, NEMOH

#### 3.2. Generation of the Incident Wave Field for Irregular Waves

#### 3.3. Generation of the Perturbed Wave Field for Irregular Waves

#### 3.4. Generation of the Total Wave Field for Irregular Waves

## 4. Validation Strategy of the Coupling Methodology between the Wave Propagation Model, MILDwave, and the Wave–Structure Interaction Solver, NEMOH

#### 4.1. Validation Test Cases

#### 4.1.1. WECwakes Experimental Data-Set

#### 4.1.2. “Test Case” Program

#### 4.1.3. Numerical Set-Up in the Used Models

#### 4.2. Criteria Used for the Numerical Model Validation

- ${K}_{d}$ contour plots of the entire numerical domains;
- ${K}_{d}$ cross-sections along the length of the numerical domains (parallel to the wave propagation direction);
- Contour plots of the “Relative Difference” between the obtained ${K}_{d}$ values ($R{D}_{{K}_{d}}$) defined as:$$R{D}_{{K}_{d},D}=\frac{({K}_{d,NEMOH}-{K}_{d,coupled})}{{K}_{d,NEMOH}}\xb7100\phantom{\rule{1.em}{0ex}}\%\phantom{\rule{1.em}{0ex}}(-),$$
- The Root Mean Square Error between ${K}_{d}$ values obtained using the MILDwave-NEMOH coupled model and NEMOH for the entire numerical domain ($RMS{E}_{{K}_{d},D}$):$$RMS{E}_{{K}_{d},D}=\sqrt{\frac{{\sum}_{i=1}^{G}{({K}_{d,NEMOH}-{K}_{d,coupled})}^{2}}{G}}\xb7100\phantom{\rule{1.em}{0ex}}\%\phantom{\rule{1.em}{0ex}}(-),$$

- Spectral density plots comparing the wave spectra between the MILDwave-NEMOH coupled model and the WECwakes experimental data for the 15 WGs.
- The Root Mean Square Error between the ${K}_{d}$ of the MILDwave-NEMOH coupled model and the ${K}_{d,WECwakes}$ of the WECwakes experimental data for the 15 WGs, $RMS{E}_{{K}_{d,WG}}$:$$RMS{E}_{{K}_{d,WG}}=\sqrt{\frac{{\sum}_{i=1}^{T}{({K}_{d,WECwakes}-{K}_{d,coupled})}^{2}}{C}}\xb7100\phantom{\rule{1.em}{0ex}}\%\phantom{\rule{1.em}{0ex}}(-),$$

## 5. Validation Results

#### 5.1. Sensitivity Analysis for Irregular Wave Generation

#### 5.2. Comparison between MILDwave-NEMOH Coupled model and NEMOH

#### 5.2.1. Irregular Waves with Wave Period ${T}_{p}=1.26$ s

#### 5.2.2. Comparison Summary

#### 5.3. Comparison between the MILDwave-NEMOH Coupled Model and the WECwakes Experimental Data-Set

#### 5.3.1. Test Case 6

#### 5.3.2. Comparison Summary

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WEC | Wave Energy Converter |

BEM | Boundary Element Method |

CFD | Computer Fluid Dynamics |

SPH | Smoothed Particle Hydrodynamics |

PTO | Power Take-Off |

RAO | Response Amplitude Operator |

DHI | Danish Hydraulic Institute |

WG | Wave Gauge |

RMSE | Root-Mean-Square-Error |

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**Figure 1.**Set-up of the different numerical wave basins used in MILDwave. The wave gauges (WGs) are represented by the x symbol and numbered as they appear in the WECwakes experimental data-set. (

**A**) empty numerical wave basin and layout of WGs; (

**B**) numerical wave basing with a single WEC; (

**C**) numerical wave basin with an array of five WECs (1 column, 1 × 5); (

**D**) numerical wave basin with an array of nine WECs (3 columns and 3 rows, 3 × 3).

**Figure 2.**Numerical wave spectrum ${S}_{n,M}\left(f\right)$ generated at the centre of the MILDwave numerical domain for an irregular wave with ${T}_{p}$ = 1.26 s and ${H}_{s}$ = 0.104 m and different simulations parameters: (

**a**) total simulation time, ${Q}_{tot}$; (

**b**) number of regular wave components, ${N}_{f}$ and (

**c**) grid cell size, ${d}_{x}(={d}_{y})$. ${S}_{n,M}\left(f\right)$ is compared in (

**a**–

**c**) to the theoretical wave spectrum, ${S}_{t}\left(f\right)$.

**Figure 3.**${K}_{d}$ results for an irregular wave with ${T}_{p}$ = 1.26 s and ${H}_{s}$ = 0.104 m obtained using the MILDwave-NEMOH coupled model: (

**a**) Test Case 2; (

**b**) Test Case 4 and (

**c**) Test Case 6 of Table 1. Contour levels are set at an interval of 0.05 of ${K}_{d}$ value (-). The coupling region is masked out using a white solid circle which includes the WECs (indicated by using black solid circles). Incident waves are generated from the left to the right. S1 and S2 indicate the location of cross-sections.

**Figure 4.**Relative difference (%) in ${K}_{d}$, $R{D}_{Kd}$, between the MILDwave-NEMOH coupled model and NEMOH for an irregular wave of ${T}_{p}$ = 1.26 s and ${H}_{s}$ = 0.104 m: (

**a**) Test Case 2; (

**b**) Test Case 4; and (

**c**) Test Case 6 of Table 1. Contour levels are set at an interval of 2 of relative difference in ${K}_{d}$ value (-). The coupling region is masked out using a white solid circle which includes the WECs (indicated by using black solid circles). Incident waves are generated from the left to the right.

**Figure 5.**${K}_{d}$ results for the MILDwave-NEMOH coupled model and for NEMOH along two longitudinal cross-sections S1 (left) and S2 (right) as indicated in Figure 3 for: (

**a**,

**b**) Test Case 2; (

**c**,

**d**) Test Case 4, and (

**e**,

**f**) Test Case 6. The coupling region is masked out in gray colour and includes the WECs’ cross-sections, which are indicated by black vertical areas.

**Figure 6.**Root-Mean-Square-Error (RMSE) for the ${K}_{d}$, $RMS{E}_{{K}_{d},D}$, over the entire numerical domain. Comparison between the MILDwave-NEMOH coupled model and NEMOH for all Test Cases of Table 1.

**Figure 9.**Root-Mean-Square-Error for the ${K}_{d}$ for all 15 WGs, $RMS{E}_{{K}_{d,WG}}$, of Figure 1A. Comparison between the MILDwave-NEMOH coupled model and the WECwakes experimental data-set.

**Table 1.**“Test Case” program for irregular waves, and different Wave Energy Converter (WEC) (array) configurations.

Test Case | Significant Wave | Peak Wave | Water Depth, | WEC Buoy | WEC (Array) |
---|---|---|---|---|---|

Number ♯ | Height, ${\mathit{H}}_{\mathit{s}}$ (m) | Period, ${\mathit{T}}_{\mathit{p}}$ (s) | d (m) | Motion (-) | Layout (-) |

1 | 0.104 | 1.18 | 0.700 | Damped | 1 × 1 |

2 | 0.104 | 1.26 | 0.700 | Damped | 1 × 1 |

3 | 0.104 | 1.26 | 0.700 | No motion (fixed buoy) | 1 × 5 |

4 | 0.104 | 1.26 | 0.700 | Damped | 1 × 5 |

5 | 0.104 | 1.18 | 0.700 | Damped | 3 × 3 |

6 | 0.104 | 1.26 | 0.700 | Damped | 3 × 3 |

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## Share and Cite

**MDPI and ACS Style**

Verao Fernández, G.; Stratigaki, V.; Troch, P.
Irregular Wave Validation of a Coupling Methodology for Numerical Modelling of Near and Far Field Effects of Wave Energy Converter Arrays. *Energies* **2019**, *12*, 538.
https://doi.org/10.3390/en12030538

**AMA Style**

Verao Fernández G, Stratigaki V, Troch P.
Irregular Wave Validation of a Coupling Methodology for Numerical Modelling of Near and Far Field Effects of Wave Energy Converter Arrays. *Energies*. 2019; 12(3):538.
https://doi.org/10.3390/en12030538

**Chicago/Turabian Style**

Verao Fernández, Gael, Vasiliki Stratigaki, and Peter Troch.
2019. "Irregular Wave Validation of a Coupling Methodology for Numerical Modelling of Near and Far Field Effects of Wave Energy Converter Arrays" *Energies* 12, no. 3: 538.
https://doi.org/10.3390/en12030538