# Hydrodynamic Analysis of a Multibody Wave Energy Converter in Regular Waves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model and Experimental Setup

_{j}as a harmonic function of time is expressed as follows:

^{th}WEC excited by waves with amplitude “a” angular frequency ω along the pitch direction (denoted subscript 5) can be expressed as

- n is the total number of WECs (n = 3 in the present study),
- ${J}_{M5}$ is the mass moment of inertia of the M
^{th}WEC in the pitch direction, - ${A}_{MNS}(\omega )$ is the hydrodynamic-added mass inertia of the M
^{th}WEC in the pitch direction induced by the S^{th}motion of the N^{th}WEC, - ${B}_{MNS}^{rad}(\omega )$ is the hydrodynamic radiation damping of the M
^{th}WEC in the pitch direction induced by the S^{th}motion of the N^{th}WEC, - ${B}_{M5}^{vis}$ is the viscous damping of the M
^{th}WEC in the pitch direction, - ${B}_{M5}^{PTO}$ is the power take-off (PTO) damping of the M
^{th}WEC in the pitch direction, - ${K}_{M5}$ is the hydrostatic stiffness moment of the M
^{th}WEC in the pitch direction, and - ${X}_{M5}$ is the wave excitation moment acting on the M
^{th}WEC in the pitch direction.

^{th}WEC power based on linear theory can be expressed as follows:

^{th}WEC rotor satisfying $\partial {\overline{P}}_{M5}^{opt}(\omega )/\partial {\tilde{B}}_{M5}^{PTO}\text{}=\text{}0$, $\frac{{\xi}_{M5}}{a}(\omega )$ is the pitch response amplitude operator (RAO) (from Equation (2)) and are expressed as

- The wave tank is equipped with a hydraulic piston-type wave maker at one end to generate waves and a wave absorber at the other end.
- The WEC rotates due to wave excitation around a fixed-axis shaft (y-direction) located below the waterline, and both ends of the shaft are connected to vertical rods, which are supported from the fixed frame along the wave flume.
- A ball bearing mechanism is provided for smooth rotation in the pitch motion.
- An image processing technique is used to measure the pitch motion of the WEC rotor.

## 3. Computational Fluid Dynamics

_{i}, P) and fluctuating (${u}_{i}^{\prime},{p}^{\prime}$) components, where Reynolds-averaged Navier–Stokes (RANS) equations along with the continuity are expressed as follows:

_{ij}is Kronecker delta function and $\upsilon $ is the kinematic viscosity of the fluid.

^{th}WEC rotor is expressed as follows:

^{th}WEC rotor, ${\omega}_{M}$ is the angular velocity of the M

^{th}WEC rotor, ${n}_{M}^{wec}$ is the resultant moment acting on the M

^{th}WEC rotor, and ${E}_{M}^{d}$ is the external damping moment of the M

^{th}WEC rotor.

## 4. Results and Discussion

#### 4.1. Validation

#### 4.2. Multiple WEC Rotors

#### 4.2.1. Linear BEM Results

- Each cell represents (peak frequency (rad/s), peak pitch RAO(rad/m)).
- First row: β = 0°, Second row: β = 30°, and Third row: β = 60°.

#### 4.2.2. Nonlinear CFD Results

- First row: ω = 1.09 rad/s, Second row: ω = 1.2 rad/s, and Third row: 1.32 rad/s.

#### 4.3. Optimal Time-Averaged Extracted Power

- First row: β = 0°, Second row: β = 30°, and Third row: β = 60°.

- First row: ω = 1.09 rad/s, Second row: ω = 1.2 rad/s, and Third row: 1.32 rad/s.

## 5. Conclusions

- The range of tested wave frequencies showed that the small wave-heading angle of the isolated WEC rotor increases the RAO while peak amplitudes are slightly affected in multiple WEC rotors with the linear BEM.
- The maximum reduction of the pitch RAO is focused around the peak frequency but insignificant elsewhere with the application of the PTO damping system.
- The q-factor demonstrated a constructive interaction in the range of (0.5 < ω < 1.08 and β ≤ 60°) and (1.34 < ω < 1.82 and β < 30°) for without PTO and (0.52 < ω < 1.1 and β ≤ 60°) and (1.33 < ω < 1.77 and β < 30°) for with PTO but destructive otherwise with the linear BEM.
- Investigation based on the prototype isolated and multiple WEC rotors, the pitch RAO, and the q-factor with without PTO at chosen wave frequencies demonstrated satisfactory overall consistency between the linear BEM and nonlinear CFD results, except for significant differences at 1.20 rad/s without PTO and 1.32 rad/s with PTO. These differences were qualified by nonlinear CFD simulations because the violent liquid motion at the center of the rotation leaves large slamming forces on the WEC rotor and leads to reduced and distinct responses from other wave frequencies.
- The estimated maximum time-averaged optimal power is distributed close to zero-heading angles around the resonance frequency and continuously decreases as the wave-heading angle increases.
- The linear BEM results showed overestimated extracted power compared with the nonlinear CFD due to the absence of nonlinear effects.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**(

**a**). Scale-down WEC rotor model and (

**b**) schematic view of the experimental setup equipped with the WEC rotor in the wave flume.

**Figure 4.**(

**a**) Overview of computational mesh and associated boundary conditions (

**b**) A close-up isometric view of the multiple WEC rotors.

**Figure 5.**Variation of the non-dimensional pitch RAO * obtained from the linear BEM, nonlinear CFD, and experimental results with the wave frequency (rad/s) for the isolated WEC rotor.

**Figure 6.**Comparison of nonlinear CFD and experimental results of pitch response for the isolated WEC rotor.

**Figure 7.**Pitch RAO (rad/m) of isolated and multiple WEC rotors without PTO as a function of the wave frequency (rad/s). Comparison of isolated WEC rotor and multiple WEC rotors (

**a**) 1, (

**b**) 2, and (

**c**) 3.

**Figure 8.**Pitch RAO (rad/m) of isolated and multiple WEC rotors with PTO as a function of the wave frequency (rad/s). Comparison of the isolated WEC rotor and multiple WEC rotors (

**a**) 1, (

**b**) 2, and (

**c**) 3.

**Figure 9.**q-factor as a function of the wave frequency (rad/s) for different wave-heading angles (°): (

**a**) Without PTO, and (

**b**) With PTO.

**Figure 10.**Numerical comparison of isolated WEC rotor with and without PTO for different wave frequencies: (

**a**) angular velocity and (

**b**) pitch response.

**Figure 11.**Numerical comparison of multiple WEC rotors with and without PTO for different wave frequencies: (

**a**) angular velocity of multiple WEC rotors 1 and 3, (

**b**) pitch response of multiple WEC rotors 1 and 3, (

**c**) angular velocity of multiple WEC rotor 2, and (

**d**) pitch response of multiple WEC rotor 2.

**Figure 12.**Comparison of the free surface wave elevation along the NWT for multiple WEC rotors: (

**a**) ω = 1.20 rad/s and (

**b**) ω = 1.32 rad/s.

**Figure 13.**Time averaged optimum extracted power P

_{opt}in (kW) from isolated and multiple WEC rotors as a function of the wave-heading angle (°) and wave frequency (rad/s). Comparison of isolated WEC rotor with multiple WEC rotors (

**a**) 1, (

**b**) 2, and (

**c**) 3.

Description | Prototype | Model (1:11) |
---|---|---|

Stern diameter, 2r (m) | 4 | 0.364 |

Depth of submergence, d (m) | 3.6 | 0.3275 |

Beak angle, α (°) | 60 | 60 |

Width, W (m) | 5 | 0.455 |

Total mass (kg) | 21,327.686 | 13.65 |

Pitch moment of inertia around the center of rotation, COR (kg⋅m^{2}) | 117,132.05 | 0.7479 |

Hydrostatic coefficient, K_{55}/ρg | 22.52 | 0.16398 × 10^{−2} |

Horizontal center of gravity w.r.t. COR (m) | −0.8934 | −0.0931 |

Vertical center of gravity w.r.t. COR (m) | 1.0189 | 0.0998 |

Pitch natural frequency, ω (rad/s) | 1.22 | 4.08 |

**Table 2.**Peak pitch RAO (rad/m) of isolated and multiple WEC rotors with and without the PTO damping for different wave-heading angles (°).

Isolated | Multiple | |||
---|---|---|---|---|

Rotor 1 | Rotor 2 | Rotor 3 | ||

Without PTO | 1.21, 2.248 | 1.23, 2.078 | 1.18, 1.844 | 1.23, 2.078 |

1.21, 2.096 | 1.22, 2.053 | 1.20, 1.803 | 1.19, 1.859 | |

1.21, 1.727 | 1.21, 1.690 | 1.21, 1.664 | 1.18, 1.659 | |

With PTO | 1.20, 1.130 | 1.20, 1.113 | 1.18, 1.081 | 1.20, 1.113 |

1.19, 1.055 | 1.20, 1.050 | 1.18, 1.021 | 1.18, 1.025 | |

1.17, 0.881 | 1.17, 0.880 | 1.17, 0.870 | 1.15, 0.852 |

Solution | PTO Damping | Isolated WEC Rotor | Multiple | q-factor | ||
---|---|---|---|---|---|---|

WEC Rotor 1 | WEC Rotor 2 | WEC Rotor 3 | ||||

Linear BEM | Without | 1.473 | 1.446 | 1.513 | 1.446 | 0.997 |

2.231 | 2.027 | 1.816 | 2.027 | 0.877 | ||

1.627 | 1.645 | 1.526 | 1.645 | 0.987 | ||

With | 1.344 | 1.343 | 1.359 | 1.343 | 1.003 | |

1.506 | 1.484 | 1.435 | 1.484 | 0.975 | ||

1.326 | 1.337 | 1.300 | 1.337 | 0.999 | ||

Nonlinear CFD | Without | 1.525 | 1.509 | 1.575 | 1.509 | 1.004 |

1.674 | 1.645 | 1.656 | 1.645 | 0.985 | ||

1.651 | 1.601 | 1.574 | 1.601 | 0.964 | ||

With | 1.242 | 1.299 | 1.361 | 1.299 | 1.063 | |

1.501 | 1.289 | 1.285 | 1.289 | 0.858 | ||

1.000 | 0.997 | 0.916 | 0.997 | 0.970 |

**Table 4.**Peak time-averaged optimum extracted power P

_{opt}in kW of isolated and multiple WEC rotors for different wave-heading angles (°).

Solution | Isolated WEC Rotor | Multiple | ||
---|---|---|---|---|

WEC Rotor 1 | WEC Rotor 2 | WEC Rotor 3 | ||

Linear | 32.834 | 32.375 | 29.577 | 32.375 |

28.364 | 28.303 | 26.370 | 26.350 | |

19.043 | 19.094 | 18.563 | 17.082 |

**Table 5.**Comparison of time-averaged optimum extracted power P

_{opt}in kW of isolated and multiple WEC rotors for different wave frequencies (rad/s).

Solution | Isolated | Multiple | ||
---|---|---|---|---|

WEC Rotor 1 | WEC Rotor 2 | WEC Rotor 3 | ||

Linear | 20.908 | 20.879 | 21.360 | 20.879 |

31.791 | 30.874 | 28.885 | 30.874 | |

29.800 | 30.347 | 28.658 | 30.348 | |

Nonlinear | 17.836 | 19.521 | 21.440 | 19.521 |

26.054 | 19.209 | 19.109 | 19.209 | |

11.570 | 11.494 | 9.716 | 11.494 |

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**MDPI and ACS Style**

Poguluri, S.K.; Kim, D.; Bae, Y.H. Hydrodynamic Analysis of a Multibody Wave Energy Converter in Regular Waves. *Processes* **2021**, *9*, 1233.
https://doi.org/10.3390/pr9071233

**AMA Style**

Poguluri SK, Kim D, Bae YH. Hydrodynamic Analysis of a Multibody Wave Energy Converter in Regular Waves. *Processes*. 2021; 9(7):1233.
https://doi.org/10.3390/pr9071233

**Chicago/Turabian Style**

Poguluri, Sunny Kumar, Dongeun Kim, and Yoon Hyeok Bae. 2021. "Hydrodynamic Analysis of a Multibody Wave Energy Converter in Regular Waves" *Processes* 9, no. 7: 1233.
https://doi.org/10.3390/pr9071233