# Coupling Methodology for Studying the Far Field Effects of Wave Energy Converter Arrays over a Varying Bathymetry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generic Coupling Methodology

- Any wave-structure interaction solver that describes the perturbed wave field is suitable for obtaining the input parameters for the internal wave generation boundary. Models based on potential flow theory (e.g., BEM [17,22,23]) or analytical models based on analytical calculation of coefficients or numerical models based on resolving the Navier–Stokes equations (e.g., CFD [6] or SPH) are all suitable in obtaining the perturbed wave field around the WEC array [20].
- Any wave propagation model can be used. A wave propagation boundary can be implemented in both phase-resolving and phase-averaging models.
- The methodology applies to any kind of oscillating or floating structure. In this paper, a WEC array of heaving point absorber WECs is modelled using a phase-resolving model (in order to demonstrate this numerical coupling methodology). However, it can be applied to oscillating water column WECs, overtopping WECs, wave surge WECs, floating breakwaters or platforms.

## 3. Application of the Coupling Methodology between the Wave Propagation Model, MILDwave, and the BEM Solver, NEMOH

#### 3.1. Numerical Background

#### 3.1.1. Linear Potential Flow

- The flow is inviscid.
- The flow is irrotational.
- The flow is incompressible.

#### 3.1.2. Wave Propagation Model MILDwave

#### 3.1.3. Wave-Structure Interaction Solver NEMOH

#### 3.1.4. Modelled WECs

#### 3.1.5. Wave Characteristics

#### 3.2. Coupling Methodology Implementation

#### 3.3. Experimental Data-Set Used for Numerical Validation Purposes

#### 3.4. Test Program

#### 3.4.1. Coupling Methodology Implementation for Constant Bottom Bathymetry

#### 3.4.2. Coupling Methodology Validation for Constant Bottom Bathymetry against Experimental Data

#### 3.4.3. Coupling Methodology Implementation for Varying Bathymetry

## 4. Results

#### 4.1. Coupling Methodology Implementation for Constant Bottom Bathymetry

#### 4.1.1. NEMOH Wave Field

#### 4.1.2. NEMOH-MILDwave Coupled Model Wave Field

#### 4.1.3. Comparison of the Total Wave Field Generated by NEMOH and the NEMOH-MILDwave Coupled Model

#### 4.2. Coupling Methodology Validation for Constant Bottom Bathymetry against Experimental Data

#### 4.3. Coupling Implementation for Varying Bathymetry

## 5. Discussion

## 6. 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 |

## References

- Stratigaki, V. Experimental Study and Numerical Modelling of Intra-Array Interactions and Extra-Array Effects of Wave Energy Converter Arrays. Ph.D. Thesis, Ghent University, Gent, Belgium, 2014. [Google Scholar]
- Child, B.M.F.; Venugopal, V. Optimal Configurations of wave energy devices. Ocean Eng.
**2010**, 37, 1402–1417. [Google Scholar] [CrossRef] - Garcia-Rosa, P.B.; Bacelli, G.; Ringwood, J. Control-Informed Optimal Array Layout for Wave Farms. IEEE Trans. Sustain. Energy
**2015**, 6, 575–582. [Google Scholar] [CrossRef][Green Version] - Babarit, A. On the park effect in arrays of oscillating wave energy converters. Renew. Energy
**2013**, 58, 68–78. [Google Scholar] [CrossRef] - Borgarino, B.; Babarit, A.; Ferrant, P. Impact of wave interaction effects on energy absorbtion in large arrays of Wave Energy Converters. Ocean Eng.
**2012**, 41, 79–88. [Google Scholar] [CrossRef] - Devolder, B.; Rauwoens, P.; Troch, P. Towards the numerical simulation of 5 Floating Point Absorber Wave Energy Converters installed in a line array using OpenFOAM. In Proceedings of the 12th European Wave and Tidal Energy Conference (EWTEC 2017), Cork, Ireland, 27 August–1 September 2017; pp. 739–749. [Google Scholar]
- Abanades, J.; Greaves, D.; Iglesias, G. Wave farm impact on the beach profile: A case study. Coast. Eng.
**2014**, 86, 36–44. [Google Scholar] [CrossRef] - Iglesias, G.; Carballo, R. Wave farm impact: The role of farm-to-coast distance. Renew. Energy
**2014**, 69, 375–385. [Google Scholar] [CrossRef] - Millar, D.L.; Smith, H.C.M.; Reeve, D.E. Modelling analysis of the sensitivity of shoreline change to a wave farm. Ocean Eng.
**2007**, 34, 884–901. [Google Scholar] [CrossRef] - Venugopal, V.; Smith, G. Wave Climate Investigation for an Array of Wave Power Devices. In Proceedings of the 7th European Wave and Tidal Energy Conference, Porto, Portugal, 11–13 September 2007; p. 10. [Google Scholar]
- Smith, H.C.M.; Millar, D.L.; Reeve, D.E. Generalisation of wave farm impact assessment on inshore wave climate. In Proceedings of the 7th European Wave and Tidal Energy Conference, Porto, Portugal, 11–13 September 2007. [Google Scholar]
- Beels, C.; Troch, P.; De Backer, G.; Vantorre, M.; De Rouck, J. Numerical implementation and sensitivity analysis of a wave energy converter in a time-dependent mild-slope equation model. Coast. Eng.
**2010**, 57, 471–492. [Google Scholar] [CrossRef] - Folley, M.; Babarit, A.; Child, B.; Forehand, D.; O’Boyle, L.; Silverthorne, K.; Spinneken, J.; Stratigaki, V.; Troch, P. A Review of Numerical Modelling of Wave Energy Converter Arrays. In Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering (OMAE), Rio de Janeiro, Brazil, 1–6 July 2012. [Google Scholar]
- Lee, C.H.; Newman, J.N. WAMIT User Manual, Versions 6.4, 6.4 PC, 6.3, 6.3S-PC; WAMIT, Inc.: Chestnut Hill, MA, USA, 2006. [Google Scholar]
- Babarit, A.; Delhommeau, G. Theoretical and numerical aspects of the open source BEM solver NEMOH. In Proceedings of the 11th European Wave and Tidal Energy Conference, Nantes, France, 6–11 September 2015. [Google Scholar]
- Balitsky, P.; Verao Fernandez, G.; Stratigaki, V.; Troch, P. Coupling methodology for modelling the near-field and far-field effects of a Wave Energy Converter. In Proceedings of the ASME 36th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2017), Trondheim, Norway, 25–30 June 2017. [Google Scholar]
- Verbrugghe, T.; Stratigaki, V.; Troch, P.; Rabussier, R.; Kortenhaus, A. A comparison study of a generic coupling methodology for modeling wake effects of wave energy converter arrays. Energies
**2017**, 10, 1697. [Google Scholar] [CrossRef] - Charrayre, F.; Peyrard, C.; Benoit, M.; Babarit, A. A Coupled Methodology for Wave-Body Interactions at the Scale of a Farm of Wave Energy Converters Including Irregular Bathymetry. In Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco, CA, USA, 8–13 June 2014. [Google Scholar]
- Tomey-Bozo, N.; Murphy, J.; Lewis, T.; Troch, P.; Thomas, G. Flap type wave energy converter modelling into a time-dependent mild-slope equation model. In Proceedings of the 2nd International Conference on Offshore Renewable Energies, Lisbon, Portugal, 24–26 October 2016; pp. 277–284. [Google Scholar]
- Verbrugghe, T.; Devolder, B.; Dominguez, J.; Kortenhaus, A.; Troch, P. Feasibility study of applying SPH in a coupled simulation tool for wave energy converter arrays. In Proceedings of the 12th European Wave and Tidal Energy Conference (EWTEC 2017), Cork, Ireland, 27 August–1 September 2017; pp. 679–689. [Google Scholar]
- Troch, P. MILDwave—A Numerical Model for Propagation and Transformation of Linear Water Waves; Technical Report; Department of Civil Engineering, Ghent University: Ghent, Belgium, 1998. [Google Scholar]
- Alves, M. Wave Energy Converter modelling techniques based on linear hydrodynamic theory. Numer. Model. Wave Energy Convert.
**2016**, 1, 11–65. [Google Scholar] - Verbrugghe, T.; Troch, P.; Kortenhaus, A.; Stratigaki, V.; Engsig-Karup, A.P. Development of a numerical modelling tool for combined near field and far field wave transformations using a coupling of potential flow solvers. In Proceedings of the 2nd International Conference on Renewable Energies Offshore, Lisbon, Portugal, 24–26 October 2016. [Google Scholar]
- Stratigaki, V.; Troch, P.; Stallard, T.; Forehand, D.; Kofoed, J.P.; Folley, M.; Benoit, M.; Babarit, A.; Kirkegaard, J. Wave Basin Experiments with Large Wave Energy Converter Arrays to Study Interactions between the Converters and Effects on Other Users. Energies
**2014**, 7, 701–734. [Google Scholar] [CrossRef][Green Version] - Stratigaki, V.; Troch, P.; Stallard, T.; Forehand, D.; Folley, M.; Kofoed, J.P.; Benoit, M.; Babarit, A.; Vantorre, M.; Kirkegaard, J. Sea-state modification and heaving float interaction factors from physical modelling of arrays of wave energy converters. J. Renew. Sustain. Energy
**2015**, 7, 061705. [Google Scholar] [CrossRef] - Penalba, M.; Touzón, I.; Lopez-Mendia, J.; Nava, V. A numerical study on the hydrodynamic impact of device slenderness and array size in wave energy farms in realistic wave climates. Ocean Eng.
**2017**, 142, 224–232. [Google Scholar] [CrossRef] - Troch, P.; Stratigaki, V. Phase-Resolving Wave Propagation Array Models. Numer. Model. Wave Energy Convert.
**2016**, 10, 191–216. [Google Scholar] - Radder, A.C.; Dingemans, M.W. Canonical equations for almost periodic, weakly nonlinear gravity waves. Wave Motion
**1985**, 7, 473–485. [Google Scholar] [CrossRef] - Beels, C.; Troch, P.; Kofoed, J.P.; Frigaard, P.; Kringelum, J.V.; Kromann, P.C.; Donovan, M.H.; De Rouck, J.; De Backer, G. A methodology for production and cost assessment of a farm of wave energy converters. Renew. Energy
**2011**, 36, 3402–3416. [Google Scholar] [CrossRef] - Stratigaki, V.; Troch, P.; Baelus, L.; Keppens, Y.U. Introducing wave generation by wind in a mild-slope wave propagation model MILDwave, to investigate the wake effects in the lee of a farm of wave energy converters. In Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering, Rotterdam, The Netherlands, 19–24 June 2011. [Google Scholar]
- Brorsen, M.; Helm-Petersen, J. On the Reflection of Short-Crested Waves in Numerical Models. In Proceedings of the 26th International Conference on Coastal Engineering, Copenhagen, Denmark, 22–26 June 1998; pp. 394–407. [Google Scholar]
- National Renewable Energy Laboratory. Marine and Hydrokinetic Technology Database. Available online: https://openei.org/wiki/Marine_and_Hydrokinetic_Technology_Database_759 (accessed on 1 June 2018).
- Faltinsen, O.M. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1990; p. 328. [Google Scholar]
- Sismani, G.; Babarit, A.; Loukogeorgaki, E. Impact of Fixed Bottom Offshore Wind Farms on the Surrounding Wave Field. Int. J. Offshore Polar Eng.
**2017**, 27, 357–365. [Google Scholar] [CrossRef] - Devolder, B.; Rauwoens, P.; Troch, P. Numerical simulation of a single floating point absorber wave energy converter using OpenFOAM. In Proceedings of the 2nd International Conference on Renewable energies Offshore, Lisbon, Portugal, 24–26 October 2016; pp. 197–205. [Google Scholar]
- Devolder, B.; Stratigaki, V.; Troch, P.; Rauwoens, P. CFD simulations of floating point absorber wave energy converter arrays subjected to regular waves. Energies
**2018**, 11, 641. [Google Scholar] [CrossRef] - Child, B.M.F. On the Configuration of Arrays of Floating Wave Energy Converters. Ph.D. Thesis, The University of Edinburgh, Edinburgh, UK, 2011. [Google Scholar]

**Figure 1.**Visual representation of the near field effects between neighboring oscillating WECs (represented by solid circles) in a WEC array under incident waves [1].

**Figure 2.**Flow chart of the generic coupling methodology between a wave-structure interaction solver and a wave propagation model.

**Figure 3.**Schematic of a one-way coupling (

**left**) and two-way coupling (

**right**). In the inner model domain, the motions of the studied structure(s)/WEC(s) are solved.

**Figure 4.**Plane (Top) view of the WEC array layout for nine heaving buoys. $\lambda $ indicates the direction of wave propagation.

**Figure 5.**Sketch of the incident wave propagation (

**left**) and perturbed wave propagation (

**right**) in MILDwave. The black line corresponds to the wave generation line, the black circle corresponds to the circular wave generation boundary and the grey areas correspond to absorption zones (sponge layers) down-wave, up-wave and along the sides of the numerical domain.

**Figure 7.**Plan view of the WECwakes experimental set-up in the DHI wave basin as a 5 × 5 rectilinear array. The red crosses indicate the position of all the wave gauges installed in the DHI wave basin during the experiments and the black circles indicate the location of the different WEC units. The wave paddles are denoted by the red hatched area at the bottom of the figure while the black hatched area at the top of the figure represents the installed absorbing beach. Two guiding walls were installed at the sides of the basin, denoted in blue lines [1].

**Figure 8.**Set up of the WEC array layout used for the comparison between the coupled model and the experimental tests. WECs are represented by • and wave gauges (WGs) by x. The WG are numbered as they appear in the WECwakes experimental data set. The direction of wave propagation, indicated by $\lambda $, is from left to right.

**Figure 9.**Depth view showing the location of the 9-WEC array. x–z plane (side) profile. The coastline is located at the right side of the figure. The 9-WEC array is located at the center of the domain.

**Figure 10.**NEMOH results of ${K}_{D}$ for a 9-WEC array for regular waves (

**top**) with T = 6 s (

**left**), T = 8 s (

**middle**) and T = 10 s (

**right**) and for irregular waves (

**bottom**) with ${T}_{p}$ = 6 s (

**left**), ${T}_{p}$ = 8 s (

**middle**) and ${T}_{p}$ = 10 s (

**right**). Contour levels are set at an interval 0.04 m. The white solid circles indicate the location of the WECs

**Figure 11.**NEMOH-MILDwave couple model ${K}_{D}$ values of a 9-WEC array for regular waves with T = 6 s, T = 8 s and T = 10 s and for irregular waves with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s. Contour levels are set at an interval 0.04 m. The water depth is 40 m. Waves are propagating from left to right. The coupling region is masked out using a white circle and includes the WECs. The NEMOH numerical domain is limited by black square.

**Figure 12.**Relative difference (%) in ${K}_{D}$ between NEMOH-MILDwave coupled model and NEMOH. Regular waves (

**top**) and Irregular waves (

**bottom**). For regular waves with T = 6 s, T = 8 s and T = 10 s and for irregular waves with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s. The coupling region is masked out using a white circle.

**Figure 13.**Cross-section S1 of the ${K}_{D}$ for a 9-WEC array at y = 0 m for regular waves (

**left**) and irregular waves (

**right**) for regular waves with T = 6 s, T = 8 s and T = 10 s and for irregular waves with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s. The coupling region is masked out in gray between two vertical black lines.

**Figure 14.**Cross-section S2 of the ${K}_{D}$ for a 9-WEC array at y = 200 m for regular waves (

**left**) and irregular waves (

**right**) for regular waves with T = 6 s, T = 8 s and T = 10 s and for irregular waves with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s.

**Figure 15.**Surface elevations $\eta $ for the NEMOH-MILDwave coupled model and the WECwakes experimental data for a total of 15 wave gauges shown in Figure 8.

**Figure 16.**Root Mean Square Error (RMSE) values for the free surface elevation $\eta $ for the 15 wave gauges analyzed from the data set (see Figure 8).

**Figure 17.**NEMOH-MILDwave couple model ${K}_{D}$ values of a 9-WEC array for regular waves (

**left**) with T = 6 s, T = 8 s and T = 10 s and for irregular waves (

**right**) with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s with a slopping bathymetry. Contour levels are set at an interval 0.04 m. Waves are propagating from left to right. The coupling region is masked out with a white circle.

**Figure 18.**Cross-section S1 of the ${K}_{D}$ for a 9-WEC array at y = 0 m for regular waves (

**left**) with T = 6 s, T = 8 s and T = 10 s and for irregular waves (

**right**) with ${T}_{p}$ = 6 s, ${T}_{p}$ = 8 s and ${T}_{p}$ = 10 s at a constant depth and a sloping bathymetry. The coupling region is masked out in gray between two vertical black lines.

Regular Waves | |||

Case Name | T(s) | H(m) | $\mathit{\theta}$= 0${}^{\mathbf{\circ}}$ |

A | 6 | 2 | 0 |

B | 8 | 2 | 0 |

C | 10 | 2 | 0 |

Irregular Waves | |||

Case Name | ${\mathit{T}}_{\mathit{p}}$(s) | ${\mathit{H}}_{\mathit{s}}$(m) | $\mathit{\theta}$= 0${}^{\mathbf{\circ}}$ |

D | 6 | 2 | 0 |

E | 8 | 2 | 0 |

F | 10 | 2 | 0 |

Test Number | Numerical Models | Wave Type | H (m) | T (s) | Water Depth d (m) |
---|---|---|---|---|---|

1 | NEMOH | REG | 2 | 6 | 40 |

2 | NEMOH | REG | 2 | 8 | 40 |

3 | NEMOH | REG | 2 | 10 | 40 |

4 | NEMOH-MILDwave | REG | 2 | 6 | 40 |

5 | NEMOH-MILDwave | REG | 2 | 8 | 40 |

6 | NEMOH-MILDwave | REG | 2 | 10 | 40 |

7 | NEMOH | IRREG | 2 | 6 | 40 |

8 | NEMOH | IRREG | 2 | 8 | 40 |

9 | NEMOH | IRREG | 2 | 10 | 40 |

10 | NEMOH-MILDwave | IRREG | 2 | 6 | 40 |

11 | NEMOH-MILDwave | IRREG | 2 | 8 | 40 |

12 | NEMOH-MILDwave | IRREG | 2 | 10 | 40 |

Test Number | Numerical Models | Wave Type | H (m) | T (s) | Water Depth d (m) |
---|---|---|---|---|---|

13 | NEMOH-MILDwave | REG | 2 | 6 | VAR |

14 | NEMOH-MILDwave | REG | 2 | 8 | VAR |

15 | NEMOH-MILDwave | REG | 2 | 10 | VAR |

16 | NEMOH-MILDwave | IRREG | 2 | 6 | VAR |

17 | NEMOH-MILDwave | IRREG | 2 | 8 | VAR |

18 | NEMOH-MILDwave | IRREG | 2 | 10 | VAR |

© 2018 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/).

## Share and Cite

**MDPI and ACS Style**

Verao Fernandez, G.; Balitsky, P.; Stratigaki, V.; Troch, P. Coupling Methodology for Studying the Far Field Effects of Wave Energy Converter Arrays over a Varying Bathymetry. *Energies* **2018**, *11*, 2899.
https://doi.org/10.3390/en11112899

**AMA Style**

Verao Fernandez G, Balitsky P, Stratigaki V, Troch P. Coupling Methodology for Studying the Far Field Effects of Wave Energy Converter Arrays over a Varying Bathymetry. *Energies*. 2018; 11(11):2899.
https://doi.org/10.3390/en11112899

**Chicago/Turabian Style**

Verao Fernandez, Gael, Philip Balitsky, Vasiliki Stratigaki, and Peter Troch. 2018. "Coupling Methodology for Studying the Far Field Effects of Wave Energy Converter Arrays over a Varying Bathymetry" *Energies* 11, no. 11: 2899.
https://doi.org/10.3390/en11112899