# Hydrodynamic Investigation of an Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater

^{*}

## Abstract

**:**

## 1. Introduction

_{r}, transmission coefficient K

_{t}, response amplitude operator (RAO) ξ in heave mode, and CWR η can then be derived. Note that the equation of K

_{r}

^{2}+ K

_{t}

^{2}+ η = 1 is satisfied based on the rule of energy conservation.

## 2. Formulas

_{1}is situated in the water with a uniform depth h

_{1}. Similar to the description in [18], a 2-dimensional Cartesian coordinate (o-xz) system is employed, and the center of origin is located at the cross-point of the still water plane and medial axis of the breakwater. Correspondingly, the mass term and stiffness term of the breakwater in heave mode can be expressed as M (=2ρad

_{1}) and K (=2ρga), where ρ denotes the density of water, and g represents the gravitational acceleration. The structure is subjected to a train of regular waves traveling in the positive x-direction. A is the incident wave amplitude, which is the maximum distance of a water particle from its equilibrium position during a period, and L is the wavelength, which is the distance that the wave travels during a wave period. The structure is assumed to respond only in heave mode.

_{1}, Ω

_{2}, and Ω

_{3}. The fluid motion in the whole domain can be described by the velocity potential:

_{I}is the incident velocity potential, Φ

_{D}denotes the diffraction potential, and Φ

_{R}represents the radiation potential due to heave motion.

^{2}= gk tanh (kh

_{1}). Correspondingly, the wavelength L = 2π/k.

_{R}and the angular frequency ω; thus, the radiation potential Φ

_{R}can be written as:

_{R}, the governing equation is Laplace equation, and its boundary conditions can be written as follows:

_{z}, added mass μ, and radiation damping coefficient λ in heave mode can then be computed using the following expressions, respectively:

_{0}is the bottom area of the structure, Re and Im denote the real and imaginary parts of a complex number, and n

_{z}is the unit normal vector along the positive z-axis. The detailed expressions of F

_{z}, μ, and λ can be found in [18].

_{PTO}denotes the PTO damping. For λ

_{PTO}= 0 and λ

_{PTO}= ∞, free heave-motion and fixed types, can be defined respectively. The optimal PTO damping can be defined as ${\lambda}_{\mathrm{optimal}}=\sqrt{{\left(K/\omega -\omega \left(M+\mu \right)\right)}^{2}+{\lambda}^{2}}$ according to [19]. Correspondingly, the heave RAO is defined as ξ = ζ/A.

_{PTO}can be written as (see [19]):

_{capture}/P

_{incident}.

_{r}and transmission coefficient K

_{t}can be written as:

## 3. Dimensional Analysis

_{1}, incident wave amplitude A, breadth B, draft d

_{1}, PTO damping λ

_{PTO}and heave response amplitude in heave ζ. The objective parameter includes the outputted power P

_{capture}, reflected wave amplitude A

_{r}and transmitted wave amplitude A

_{t}. The incident wave amplitude, water depth, water density and gravitational acceleration are fixed parameters in this analysis. P

_{ref}denotes an arbitrary reference power level and λ

_{ref}an arbitrary reference damping level. According to the Buckingham's theorem [20], the dimensionless variables can be determined (details see Table 1). Here, the incident wave power (i.e., P

_{incident}) was chosen as P

_{ref}and the optimal PTO damping λ

_{optimal}as λ

_{ref}[21]. Thus, the dimensionless outputted power can be expressed as the CWR η (=P

_{capture}/P

_{incident}). The reflected wave and the transmitted wave are dimensionalized by the incident wave amplitude A. Correspondingly, the dimensionless reflected wave and transmitted wave can be expressed as the reflection coefficient K

_{r}(=A

_{r}/A) and the transmission coefficient K

_{t}(=A

_{t}/A). Then, we can write the CWR, reflection coefficient and transmission coefficient as function of the dimensionless wavenumber kh

_{1}, relative breadth B/h

_{1}, relative draft d

_{1}/h

_{1}and relative PTO damping λ

_{PTO}/λ

_{optimal}.

## 4. Results and Discussion

#### 4.1. Validation

_{r}and transmission coefficient K

_{t}is considered. Figure 2 shows the variations of K

_{r}and K

_{t}with the dimensionless wavenumber kh

_{1}obtained by the present analytical model and the corresponding numerical results [22]. The geometrical parameters are a = h

_{1}, d

_{1}= 0.5h

_{1}, and h

_{1}=1 m; λ

_{PTO}= 10000 λ

_{optimal}is used to solve for the motion equation in the present study, in order to have the pontoon fixed as in [22]. The incident wave amplitude is A = 0.1 m. The maximum difference between the present and reference results for the reflection and transmission coefficients are 5% and 2.5%, respectively. As shown, a good agreement can be achieved.

_{1}and d

_{1}= 0.2h

_{1}. Since the breakwater is constrained to heave motion, but no PTO damping was considered in [23], the value chosen for the PTO damping λ

_{PTO}is 0 in the present study. Figure 3 shows the comparison of the heave RAO obtained using the present approach with the results obtained in [23].The maximum difference between the present and reference results for the heave RAO is 4.5%. It can be seen that a good agreement can be obtained.

#### 4.2. Comparison of the Different Breakwater Systems

_{r}, transmission coefficient K

_{t}, and CWR η for the proposed integrated device with the optimal PTO damping λ

_{optimal}are critical in determining the frequency region in which the acceptable wave attenuation performance and the efficient wave energy conversion can be obtained.

_{1}= 2.5 m, B = 8 m, and h

_{1}= 10 m (i.e., B/h

_{1}= 0.8 and d

_{1}/h

_{1}= 0.25). The incident wave amplitude is A = 1.0 m. Figure 4, Figure 5 and Figure 6 present a comparison of the three cases with respect to the reflection coefficient K

_{r}, transmission coefficient K

_{t}, and heave RAO ξ, respectively.

_{t}, K

_{r}, η, and K

_{t}

^{2}+ K

_{r}

^{2}+ η with the dimensionless wavenumber kh

_{1}for the breakwater with the optimal PTO damping. The curve of η exhibits a parabolic trend and reaches the maximum (i.e., η

_{max}= 50%) at resonance. Interestingly, there exists a cross-point for the three curves at which η is 50%, and both K

_{r}and K

_{t}are 0.5. This observation indicates that 25% of the incident wave energy is reflected toward the left, 25% is transmitted toward the right, and the remaining 50% is absorbed. This is a consequence of wave energy theory (see [19] (p. 198)). The condition K

_{t}< 0.5 shall be satisfied for a qualified breakwater [25]. Therefore, the ideal frequency region with a lower threshold corresponding to the natural frequency is of interest for the integrated system with the optimal PTO damping. Since the study is conducted under the context of small amplitude assumption within linear potential theory neglecting viscous and nonlinear effects, the heave RAO of the device may not be totally in accordance with the corresponding experimental results. From the literature, it can be seen that the whole variation trends of the hydrodynamic coefficients corresponding to analytical and experimental results are similar [23]. Thus, the analytical results may have directive significance for the practical engineering.

#### 4.3. Effect of the Relative Breadth B/h_{1}

_{1}) and dimensionless draft (i.e., relative draft d

_{1}/h

_{1}) on the hydrodynamic properties of the system are conducted, respectively. Figure 8, Figure 9, Figure 10 and Figure 11 show the variations of the reflection coefficient K

_{r}, transmission coefficient K

_{t}, CWR η and heave RAO ξ with the dimensionless wavenumber kh

_{1}for different relative breadths B/h

_{1}(=0.2, 0.5, 0.8, 1.1 and 1.4). The other geometrical parameters are d

_{1}= 2.5 m and h

_{1}= 10 m (i.e., d

_{1}/h

_{1}= 0.25). The incident wave amplitude A is 1.0 m. Optimal damping is used in this subsection. From Figure 8 and Figure 9, it can be seen that, for the pontoon-type floating breakwater, the wider the breadth of the breakwater, the more effective is the wave barrier [17,25]. For the CWR, the maximum CWR does not vary with the relative draft. The maximum heave RAO obviously increases with the decreasing of the relative breadth. The dimensionless wavenumber kh

_{1}corresponding to the maximum heave RAO (or the CWR) decreases with the increasing of the relative breadth. It is due to the fact that the natural frequencies of the pontoon decrease with the increasing of the relative breadth.

#### 4.4. Effect of the Relative Draft d_{1}/h_{1}

_{r}, transmission coefficient K

_{t}, CWR η and heave RAO ξ with the dimensionless wavenumber kh

_{1}for different relative drafts d

_{1}/h

_{1}= 0.05, 0.15, 0.25, 0.35 and 0.45. The other geometrical parameters are B = 8 m and h

_{1}= 10 m (i.e., B/h

_{1}= 0.8). The incident wave amplitude A is 1.0 m. From Figure 12 and Figure 13, it can be seen that, the deeper the draft of the breakwater, the more effective is the wave barrier. For the CWR, the effective frequency bandwidth (η > 20%) narrows with an increase in the draft. However, the maximum CWR does not vary with the relative draft. The maximum heave RAO increases with the increasing of the relative draft. Note that, since the natural frequencies of the pontoon decreases with the increasing of relative draft, the kh

_{1}corresponding to the maximum CWR and heave RAO decreases accordingly.

#### 4.5. Effect of the PTO Damping

_{r}, transmission coefficient K

_{t}, and CWR η are considered. The geometrical parameters are d

_{1}= 2.5 m, B = 8 m and h

_{1}= 10 m (i.e., d

_{1}/h

_{1}= 0.25, B/h

_{1}= 0.8). The incident wave amplitude A is 1.0 m. The values selected for the tested PTO dampings are λ

_{PTO}= 0.8λ

_{optimal}, 1.0λ

_{optimal}, 1.5λ

_{optimal}, 2.0λ

_{optimal}, 5.0λ

_{optimal}, and λ

_{PTO}= 10000λ

_{optimal}(i.e., the case of the fixed breakwater).

_{r}with the dimensionless wavenumber kh

_{1}. As observed, the reflection coefficient increases with an increase in the PTO damping. Figure 17 and Figure 18 show the variations of the transmission coefficient and heave RAO against kh

_{1}. With an increase in the PTO damping, the heave RAO decreases (referring to Figure 18); the transmission coefficient increases in the lower frequencies and, differently, the trend of decreasing firstly and then increasing was found in the middle frequency region (i.e., 1.3 < kh

_{1}< 2.7). These findings indicate that the wave attenuation performance can be superior to the fixed breakwater by proper adjustment of the PTO damping. Figure 19 shows the variations of the CWR η with kh

_{1}. Since the natural frequency ω

_{nat}of the pontoon in heave mode can be expressed as ω

_{nat}= $\sqrt{\frac{K}{M+\mu}}$, changes in the PTO damping do not affect the natural frequency of the system [26]. Thus, the locations (i.e., kh

_{1}) of the CWR peak value are similar for the different cases. With an increase in the PTO damping, the CWR first increases and then decreases. Notably, the CWR corresponding to the PTO damping of λ

_{PTO}= 1.5λ

_{optimal}(or λ

_{PTO}= 2λ

_{optimal}) is only slightly inferior to the case with the optimal PTO damping in the range of 1.3 < kh

_{1}< 2.7; under this condition, the transmission coefficient of the former is superior to the latter. Specifically, considering the conditions K

_{t}< 0.5 and η > 20%, the available frequency region is 1.925 < kh

_{1}< 3.075 when λ

_{PTO}= 1λ

_{optimal}; 1.723 < kh

_{1}< 3.02 when λ

_{PTO}= 1.5λ

_{optimal}, and 1.625 < kh

_{1}< 2.92 when λ

_{PTO}= 2λ

_{optimal}. That is, the effective frequency bandwidth is broadened when λ

_{PTO}= 1.5–2λ

_{optimal}.

## 5. Conclusions

- (1)
- Compared with that of the free heave-motion breakwater, the wave attenuation performance of the breakwater is improved for the proposed integrated system.
- (2)
- For the system with the optimal PTO damping, the low threshold of the practical frequency region corresponds to the natural frequency.
- (3)
- With a decrease in the heave RAO of the breakwater, the transmission coefficient increases in the lower-frequency region, although a decreasing trend is initially observed, followed by an increasing trend in the middle-frequency region.
- (4)
- Due to the changing of the natural frequency, the effect of the relative breadth B/h
_{1}and relative draft d_{1}/h_{1}of the pontoon affect the performance of the system significantly. This shall be paid attention while such a system is designed. - (5)
- The breakwater with the PTO damping of λ
_{PTO}= 1.5–2λ_{optimal}may give a broader frequency bandwidth with K_{t}< 0.5 and η > 20%. Fortunately, the transmission coefficient corresponding to the case with λ_{PTO}= 2λ_{optimal}is slightly superior to that of the fixed breakwater. - (6)
- The proposed system is theoretically proved to produce power effectively and, at the same time, the function of coastal protection can be comparable to that of the fixed breakwater.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Sketch of the floating structure with the power take-off (PTO) system (the structure restrained with the vertical pile moves in heave and the PTO system is used to capture wave energy, detailed description see [17]).

**Figure 2.**Comparison of transmission coefficient K

_{r}and reflection coefficient K

_{t}obtained by the present approach with the results of [22].

**Figure 3.**Comparison of the heave response amplitude operator (RAO) obtained using the present approach with the results of Isaacson et al. [23]. (B/L represents the dimensionless wavelength).

**Figure 4.**Variations of the reflection coefficient K

_{r}, transmission coefficient K

_{t}, capture width ratio (CWR) η, and K

_{t}

^{2}+ K

_{r}

^{2}+ η with the dimensionless wavenumber kh

_{1}for the breakwater with the optimal PTO damping.

**Figure 5.**Variations of the reflection coefficient K

_{r}with the dimensionless wavenumber kh

_{1}for cases 1–3.

**Figure 6.**Variations of the transmission coefficient K

_{t}with the dimensionless wavenumber kh

_{1}for cases 1–3.

**Figure 8.**Variations of the reflection coefficient K

_{r}with the dimensionless wavenumber kh

_{1}for cases of relative draft d

_{1}/h

_{1}= 0.25 and PTO damping λ

_{PTO}= λ

_{optimal}(λ

_{optimal}refers to the optimal PTO damping).

**Figure 9.**Variations of the transmission coefficient K

_{t}with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and λ

_{PTO}= λ

_{optimal}.

**Figure 10.**Variations of the CWR η with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and λ

_{PTO}= λ

_{optimal}.

**Figure 11.**Variations of the heave RAO ξ with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and λ

_{PTO}= λ

_{optimal}.

**Figure 12.**Variations of the reflection coefficient K

_{r}with the dimensionless wavenumber kh

_{1}for cases of relative breadth B/h

_{1}= 0.8 and λ

_{PTO}= λ

_{optimal}.

**Figure 13.**Variations of the transmission coefficient K

_{t}with the dimensionless wavenumber kh

_{1}for cases of B/h

_{1}= 0.8 and λ

_{PTO}= λ

_{optimal}.

**Figure 14.**Variations of the CWR η with the dimensionless wavenumber kh

_{1}for cases of B/h

_{1}= 0.8 and λ

_{PTO}= λ

_{optimal}.

**Figure 15.**Variations of the heave RAO ξ with the dimensionless wavenumber kh

_{1}for cases of B/h

_{1}= 0.8 and λ

_{PTO}= λ

_{optimal}.

**Figure 16.**Variations of the reflection coefficient K

_{r}with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and B/h

_{1}= 0.8.

**Figure 17.**Variations of the transmission coefficient K

_{t}with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and B/h

_{1}= 0.8.

**Figure 18.**Variations of the heave RAO ξ with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and B/h

_{1}= 0.8.

**Figure 19.**Variations of the CWR η with the dimensionless wavenumber kh

_{1}for cases of d

_{1}/h

_{1}= 0.25 and B/h

_{1}= 0.8.

Dimensional Variables | Physical Unit | Nondimensional Variables |
---|---|---|

Water density, ρ | kg·m^{−3} | - |

Gravitational acceleration, g | m·s^{−2} | - |

Water depth, h_{1} | m | - |

Incident wave amplitude, A | m | - |

Wavenumber, k | m^{–1} | π_{1} = kh_{1} |

Breadth, B | m | π_{2} = B/h_{1} |

Draft, d_{1} | m | π_{3} = d_{1}/h_{1} |

PTO damping, λ_{PTO} | kg·s^{−1} | π_{4} = λ_{PTO}/λ_{ref} |

Response amplitude in heave, ζ | m | π_{5} = ζ/A |

Outputted power, P_{capture} | kg·m^{2}·s^{−3} | π_{a} = P_{capture}/P_{ref} |

Reflected wave amplitude, A_{r} | m | π_{b} = A_{r}/A |

Transmitted wave amplitude, A_{t}, | m | π_{c} = A_{t}/A |

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## Share and Cite

**MDPI and ACS Style**

Zhao, X.; Ning, D.; Zhang, C.; Kang, H.
Hydrodynamic Investigation of an Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater. *Energies* **2017**, *10*, 712.
https://doi.org/10.3390/en10050712

**AMA Style**

Zhao X, Ning D, Zhang C, Kang H.
Hydrodynamic Investigation of an Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater. *Energies*. 2017; 10(5):712.
https://doi.org/10.3390/en10050712

**Chicago/Turabian Style**

Zhao, Xuanlie, Dezhi Ning, Chongwei Zhang, and Haigui Kang.
2017. "Hydrodynamic Investigation of an Oscillating Buoy Wave Energy Converter Integrated into a Pile-Restrained Floating Breakwater" *Energies* 10, no. 5: 712.
https://doi.org/10.3390/en10050712