# Wake Effect Assessment in Long- and Short-Crested Seas of Heaving-Point Absorber and Oscillating Wave Surge WEC Arrays

^{*}

## Abstract

**:**

## 1. Introduction

## 2. MILDwave-NEMOH Coupled Model

#### 2.1. Basis of the MILDwave-NEMOH Coupled Model

- A first simulation is performed in MILDwave to obtain the incident wave field in the time-domain, without any floating structure in the numerical basin. The wave characteristics at the coupling location are computed and used as input values for NEMOH.
- A second simulation is performed in NEMOH to calculate the perturbed wave field around the floating structure at the coupling location in the frequency-domain.
- A third simulation is performed in MILDwave to obtain the perturbed wave field in the time-domain. The perturbed wave field from NEMOH is transformed from the frequency-domain to the time-domain and coupled into MILDwave by prescribing an internal wave generation boundary.
- Finally, the total wave field is obtained as the combination of the incident and perturbed wave fields calculated in MILDwave in the time-domain.

#### 2.2. The Wave Propagation Model MILDwave

#### 2.3. The Wave-Structure Interaction Solver, NEMOH

- The flow is inviscid.
- The flow is irrotational.
- The fluid is incompressible.
- The motion amplitudes of the modelled floating bodies are much smaller than the wavelength.
- The sea bottom is flat.

#### 2.4. Generation of the Incident Wave Field

#### 2.5. Calculation of the Perturbed Wave Field

#### 2.6. Calculation of the Total Wave Field

## 3. Numerical Framework

#### 3.1. Modelled WECs and Array Layout

- Heaving Cylindrical Wave Energy Converter (HCWEC): is a disc shaped heaving buoy WEC with a diameter, ∅, of 20.0 m, a height, ${h}_{HC}$, of 4.0 m, and a draft, ${d}_{HC}$, of 2.0 m. HCWECs are designed to be deployed at water depths of around 30.0 m.
- Oscillating Wave Surge Wave Energy Converter (OSWEC): the second WEC chosen is a bottom-fixed pitching flap driven by the surge motion of the waves. OSWECs are designed to be deployed in shallow water at depths of 10.0–20.0 m. The simulated OSWEC has dimensions of 20.0 m width, ${w}_{OS}$, 1.0 m thickness, ${t}_{OS}$, and 12.0 m height, ${h}_{OS}$. It is hinged at the seabed with pitching motion about its bottom end.

#### 3.2. Wave Conditions

#### 3.3. Numerical Set-Up

#### 3.4. Test Cases

## 4. Results

#### 4.1. ${K}_{d}$ Disturbance Coefficient for Short-Crested Irregular Waves

#### 4.2. 9-OSWEC Array

#### 4.3. 9-HCWEC Array

#### 4.4. Comparison Summary

#### 4.5. Computational Time

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WEC | Wave Energy Converter |

BEM | Boundary Element Method |

PTO | Power Take-Off |

RAO | Response Amplitude Operator |

P-M | Pierson–Moskowitz |

DOF | Degree of freedom |

HCWEC | Heaving Cylindrical Wave Energy Converter |

OSWEC | Oscillating Wave Surge Wave Energy Converter |

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**Figure 3.**Set-up of the different numerical wave basins used in MILDwave. (

**A**) Incident Wave Field numerical wave basin. (

**B**) Perturbed wave field numerical wave basin with a single WEC. The coupling region corresponds to the wave-structure interaction solver NEMOH domain defined in the perturbed wave field numerical wave basin.

**Figure 4.**Plane (top) view of all different WEC array layouts used in the numerical simulations. (

**A**) 5-HCWEC and 9-HCWEC arrays, and (

**B**) 5-OSWEC and 9-OSWEC arrays.

**Figure 5.**${K}_{d}$ disturbance coefficient results for a single HCWEC interacting with a regular wave with H = 2.0 m, T = 8.0 s and $\theta $ = ${0}^{\circ}$. The coupling region is filled using a white solid circle which includes the WEC (indicated by a black solid circle). Incident waves are generated from the left to the right.

**Figure 6.**${K}_{d}$ disturbance coefficient results for a single HCWEC interacting with a regular wave with H = 2.0 m, T = 8.0 s and $\theta $ = ${30}^{\circ}$. The coupling region is filled using a white solid circle which includes the WEC (indicated by a black solid circle). Incident waves are generated from the left to the right.

**Figure 7.**Normalized spreading function distributions ${D}_{N}(f,\theta )$ generated at the center of the MILDwave numerical domain for an incident short-crested irregular wave. ${H}_{s}$ = 2 m and ${T}_{p}$ = 8 s, and different randomly generated short-crested sea states are used with ${s}_{1}=15.8$.

**Figure 8.**${K}_{d,avg}$ disturbance coefficient results for a single HCWEC interacting with short-crested irregular waves with ${H}_{s}$ = 2.0 m, ${T}_{p}$ = 8.0 s and ${s}_{1}$ = 15.8. The coupling region is filled using a white solid circle which includes the WEC (indicated by a black solid circle). Incident waves are generated from the left to the right.

**Figure 9.**${K}_{d}$ disturbance coefficient results for a 9-OSWEC array (O2) (

**A**) and a 9-HCWEC (H2) (

**B**) array interacting with regular waves with H = 2.0 m and T = 8.0 s. The coupling region is filled using a white solid circle which includes the WEC (indicated by black solid circles). Incident waves are generated from the left to the right. S1, W1, W2, W3, and W4 indicate the locations of cross-sections.

**Figure 10.**${K}_{d}$ disturbance coefficient results for a 9-OSWEC array (O2) (

**A**) and a 9-HCWEC array (H2) (

**B**) interacting with long-crested irregular waves with ${H}_{s}$ = 2.0 m, ${T}_{p}$ = 8.0 s and ${s}_{1}$ = 0.0${}^{\circ}$. The coupling region is filled using a white solid circle which includes the WEC (indicated by black solid circles). Incident waves are generated from the left to the right. S1, W1, W2, W3, and W4 indicate the locations of cross-sections.

**Figure 11.**${K}_{d}$ disturbance coefficient results for a 9-OSWEC array (O2) (

**A**) and a 9-HCWEC array (H2) (

**B**) interacting with short-crested irregular wave with ${H}_{s}$ = 2.0 m, ${T}_{p}$ = 8.0 s and ${s}_{1}$ = 15.8. The coupling region is filled using a white solid circle which includes the WEC (indicated by black solid circles). Incident waves are generated from the left to the right. S1, W1, W2, W3, and W4 indicate the location of cross-sections.

**Figure 12.**${K}_{d}$ disturbance coefficient results along one longitudinal cross-section S1 as indicated in Figure 9, Figure 10 and Figure 11, for (

**A**), a 9-OSWEC (O2) array, and (

**B**) a 9-HCWEC array, interacting with regular, irregular long-crested, and irregular short-crested waves, respectively. The results are obtained using the MILDwave-NEMOH coupled. The coupling region is indicated using gray color and includes the WECs’ cross-sections, which are indicated by black vertical areas.

**Figure 13.**${K}_{d}$ disturbance coefficient results along four transversal cross-sections W1, W2, W3 and W4 as indicated in Figure 9, Figure 10 and Figure 11, for a 9-OSWEC (O2) and a 9-HCWEC (H2) array interacting with regular, irregular long-crested, and irregular short-crested waves, respectively.

Regular Waves | Long-Crested Irregular Waves | Short-Crested Irregular Waves |
---|---|---|

H = $2.0$ m | ${H}_{s}=2.0$ m | ${H}_{s}$ = $2.0$ m |

$T=8.0$ s | ${T}_{p}=8.0$ s | ${T}_{p}$ = $8.0$ s |

$\theta ={30.0}^{\circ}$ | $\theta ={30.0}^{\circ}$ | $\theta ={30.0}^{\circ}$ |

${l}_{MW}$ = 2000.0 m | ${l}_{MW}$ = 2000.0 m | ${l}_{MW}$ = 2000.0 m |

${w}_{MW}$ = 2000.0 m | ${w}_{MW}$ = 2000.0 m | ${w}_{MW}$ = 2000.0 m |

- | Pierson–Moskowitz Spectrum | Pierson–Moskowitz Spectrum |

- | ${N}_{f}$ = 20 | ${N}_{f}$ = 50 |

- | ${s}_{1}$ = 0.0 | ${s}_{1}$ = 15.8 |

${d}_{x}$ = L/30 = 3.0 m | ${d}_{x}$ = ${L}_{p}$/30 = 3.0 m | ${d}_{x}$ = ${L}_{p}$/30 = 3.0 m |

${d}_{y}$ = L/30 = 3.0 m | ${d}_{y}$ = ${L}_{p}$/30 = 3.0 m | ${d}_{y}$ = ${L}_{p}$/30 = 3.0 m |

${t}_{sim}$ = 600 s | ${t}_{sim}$ = 4000 s | ${t}_{sim}$ = 5000 s |

$\Delta t$ = 0.4 s | $\Delta t$ = 0.4 s | $\Delta t$ = 0.4 s |

Regular Waves | Long-Crested Irregular Waves | Short-Crested Irregular Waves |
---|---|---|

H = $2.0$ m | ${H}_{s}=2.0$ m | ${H}_{s}$ = $2.0$ m |

$T=8.0$ s | ${T}_{p}=8.0$ s | ${T}_{p}$ = $8.0$ s |

$\theta ={30.0}^{\circ}$ | $\theta ={30.0}^{\circ}$ | $\theta ={30.0}^{\circ}$ |

${l}_{NM}$ = 400.0 m | ${l}_{NM}$ = 400.0 m | ${l}_{NM}$ = 400.0 m |

${w}_{NM}$ = 400.0 m | ${w}_{NM}$ = 400.0 m | ${w}_{NM}$ = 400.0 m |

- | Pierson–Moskowitz Spectrum | Pierson–Moskowitz Spectrum |

- | ${N}_{f}$ = 20 | ${N}_{f}$ = 50 |

- | ${s}_{1}$ = 0.0 | ${s}_{1}$ = 15.8 |

${d}_{x}$ = L/30 = 3.0 m | ${d}_{x}$ = ${L}_{p}$/30 = 3.0 m | ${d}_{x}$ = ${L}_{p}$/30 = 3.0 m |

${d}_{y}$ = L/30 = 3.0 m | ${d}_{y}$ = ${L}_{p}$/30 = 3.0 m | ${d}_{y}$ = ${L}_{p}$/30 = 3.0 m |

Test Case | Wave Type | WEC (array) |
---|---|---|

1 | Regular | Single HCWEC |

2 | Regular | H1 array layout |

3 | Regular | H2 array layout |

4 | Regular | Single OSWEC |

5 | Regular | O1 array layout |

6 | Regular | O2 array layout |

7 | Irregular long-crested | Single HCWEC |

8 | Irregular long-crested | H1 array layout |

9 | Irregular long-crested | H2 array layout |

10 | Irregular long-crested | Single OSWEC |

11 | Irregular long-crested | O1 array layout |

12 | Irregular long-crested | O2 array layout |

13 | Irregular short-crested | Single HCWEC |

14 | Irregular short-crested | H1 array layout |

15 | Irregular short-crested | H2 array layout |

16 | Irregular short-crested | Single OSWEC |

17 | Irregular short-crested | O1 array layout |

18 | Irregular short-crested | O2 array layout |

P1(250,0) | P2(500,0) | P3(750,0) | P4(1000,0) | |
---|---|---|---|---|

1 HCWEC | ||||

Regular waves | −3.11 | −2.15 | −1.77 | −1.54 |

Long-crested irregular waves | −4.7 | −3.24 | −2.66 | −2.31 |

Short-crested irregular waves | −4.92 | −3.01 | −2.37 | −2.03 |

1 OSWEC | ||||

Regular waves | −7.87 | −5.74 | −4.74 | −4.15 |

Long-crested irregular waves | −7.53 | −5.28 | −4.42 | −3.90 |

Short-crested irregular waves | −5.21 | −3.39 | −2.87 | −2.85 |

5 HCWEC | ||||

Regular waves | −17.99 | −12.19 | −9.89 | −8.53 |

Long-crested irregular waves | −21.21 | −14.93 | −11.96 | −10.23 |

Short-crested irregular waves | −16.69 | −10.66 | −7.91 | −6.12 |

5 OSWEC | ||||

Regular waves | −22.12 | −20.67 | −18.87 | −17.50 |

Long-crested irregular waves | −23.52 | −16.50 | −15.08 | −14.06 |

Short-crested irregular waves | −20.37 | −13.63 | −10.42 | −8.64 |

9 HCWEC | ||||

Regular waves | −22.75 | −21.51 | −18.12 | −15.75 |

Long-crested irregular waves | −19.82 | −21.90 | −21.72 | −18.71 |

Short-crested irregular waves | −18.49 | −15.80 | −12.50 | −10.17 |

9 OSWEC | ||||

Regular waves | −39.08 | −26.15 | −22.67 | −22.48 |

Long-crested irregular waves | −30.81 | −27.83 | −22.72 | −18.90 |

Short-crested irregular waves | −26.55 | −20.28 | −16.23 | −13.59 |

Computational Time (h) | |||||
---|---|---|---|---|---|

Test Case | J, Number | NEMOH | MILDwave | MILDwave | Total |

Number | of Bodies [-] | Perturbed Wave | Incident Wave | Perturbed Wave | Computational |

Simulation | Simulation | Simulation | Time | ||

1 | 1 | 0.00 | 0.18 | 0.22 | 0.40 |

2 | 5 | 0.03 | 0.18 | 0.22 | 0.43 |

3 | 9 | 0.08 | 0.18 | 0.22 | 0.48 |

7 | 1 | 0.21 | 0.5 | 0.77 | 1.48 |

8 | 5 | 0.41 | 0.5 | 0.77 | 1.68 |

9 | 9 | 1.51 | 0.5 | 0.77 | 2.78 |

10 | 1 | 0.08 | 1.18 | 1.21 | 2.47 |

11 | 5 | 0.75 | 1.18 | 1.21 | 4.14 |

12 | 9 | 4.16 | 1.18 | 1.21 | 6.63 |

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## Share and Cite

**MDPI and ACS Style**

Verao Fernandez, G.; Stratigaki, V.; Vasarmidis, P.; Balitsky, P.; Troch, P.
Wake Effect Assessment in Long- and Short-Crested Seas of Heaving-Point Absorber and Oscillating Wave Surge WEC Arrays. *Water* **2019**, *11*, 1126.
https://doi.org/10.3390/w11061126

**AMA Style**

Verao Fernandez G, Stratigaki V, Vasarmidis P, Balitsky P, Troch P.
Wake Effect Assessment in Long- and Short-Crested Seas of Heaving-Point Absorber and Oscillating Wave Surge WEC Arrays. *Water*. 2019; 11(6):1126.
https://doi.org/10.3390/w11061126

**Chicago/Turabian Style**

Verao Fernandez, Gael, Vasiliki Stratigaki, Panagiotis Vasarmidis, Philip Balitsky, and Peter Troch.
2019. "Wake Effect Assessment in Long- and Short-Crested Seas of Heaving-Point Absorber and Oscillating Wave Surge WEC Arrays" *Water* 11, no. 6: 1126.
https://doi.org/10.3390/w11061126