# Analytical Study on an Oscillating Buoy Wave Energy Converter Integrated into a Fixed Box-Type Breakwater

Authors: Xuanlie Zhao; Dezhi Ning; Chongwei Zhang; Yingyi Liu; Haigui Kang
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

An oscillating buoy wave energy converter (WEC) integrated to an existing box-type breakwater is introduced in this study. The buoy is installed on the existing breakwater and designed to be much smaller than the breakwater in scale, aiming to reduce the construction cost of the WEC. The oscillating buoy works as a heave-type WEC in front of the breakwater towards the incident waves. A power take-off (PTO) system is installed on the topside of the breakwater to harvest the kinetic energy (in heave mode) of the floating buoy. The hydrodynamic performance of this system is studied analytically based on linear potential-flow theory. Effects of the geometrical parameters on the reflection and transmission coefficients and the capture width ratio (CWR) of the system are investigated. Results show that the maximum efficiency of the energy extraction can reach 80% or even higher. Compared with the isolated box-type breakwater, the reflection coefficient can be effectively decreased by using this oscillating buoy WEC, with unchanged transmission coefficient. Thus, the possibility of capturing the wave energy with the oscillating buoy WEC integrated into breakwaters is shown.


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## Body

## 1. Introduction

Breakwaters are commonly constructed at a distance from the coastline, aiming to reduce coastal erosion or provide safe harborage. At shallow-water deployment sites, it is usually convenient to use bottom-sitting breakwaters such as the rubble mound or vertical wall breakwaters. The breakwater structure works by either absorbing the energy of the waves that hit it or simply reflecting the inshore waves away [1]. Due to simplification in structures, the box-type breakwater is most widely applied. It works by reflecting incident waves in a large extent, so that the leeward transmitted waves can be reduced greatly. The hydrodynamic performance of such breakwater has been systematically studied [1–8].In recent years, the requirement for renewable energy increases greatly. To decrease the construction cost, some new ideas on integrating the wave energy converter (WEC) into some kind of coastal and offshore structures have been raised (Dal Ferro 2006). For example, Michailides and Angelides [9] proposed a flexible floating breakwater. Each breakwater has several hinged floating modules, through whose relative motion the wave energy can be captured. He and Huang [10] integrated the classical oscillating water column (OWC) WEC into the floating breakwater structure. Through experiments, the possibility of using OWC to mitigate the motion of the breakwater and wave transmission was shown. Chen et al. [11] considered two floating cylinders used as both WECs and floating breakwaters. The performance of the device was investigated numerically. He and Huang [12] arranged an OWC device in front of a vertical wall. Their experimental results showed that OWC structure can be served as a wave absorber for reducing wave reflection from the vertical wall. Ning et al. [13] introduced a buoy-type WEC combined with the pile-restrained floating breakwater. Experimental results showed that good wave attenuation performance and energy-conversion efficiency can be achieved, with appropriate dimensions and power take-off (PTO) damping setting. Overall, designs based on the combination of WECs and breakwaters have evident advantages. For example, the WEC and breakwaters can share the cost to a large proportion; ocean space for WEC device can be saved greatly, and the performance of breakwaters may be further improved.In this study, we will propose a more economical design by integrating the oscillating buoy WEC into the existing box-type breakwaters. The oscillating buoy WEC works as a floating box in front of the breakwater towards the incident waves. A power take-off (PTO) system is installed on the topside of the breakwater to harvest the kinetic energy of the front buoy. Thus, the energy of incident and reflected waves can be converted to the usable electricity, instead of being reflected away as the typical box-type breakwater does. Theoretically, the buoy in front of the breakwater can capture more wave energy than an isolated oscillating buoy WEC, as a result of the superposition of incident and reflected waves.At this preliminary design stage, this study will focus on the hydrodynamic performance of this system. Based on the efficiency considerations, the analytical solution of the hydrodynamic problem is firstly derived using potential-flow theory, as in Section2. Then, in Section 3, validation and parametric studies are conducted, respectively. The reflection coefficient and capture width ratio (CWR) of the system are investigated to show the effectiveness of this device. The wave transmission is also tested to check whether the buoy motion affects the intrinsic performance of the breakwater. In Section 4, the conclusions are drawn.


## 2. Mathematical Equations

As shown in Figure1, a floating buoy and a box-type breakwater are placed in the water with uniform depth h. The displacement of the front buoy is much smaller than that of the breakwater. The breadths of two structures are defined as a1 and a2, the drafts are d1 and d2, respectively, and the spacing in between is D. The mass of the buoy is M1=ρa1d1, with ρ as the water density. The buoy can only undergo a heave motion restrained by a spring system. The stiffness coefficient of the spring can be expressed as K1=ρga1, where g is the gravity acceleration. A 2D Cartesian coordinate systemO-xz is defined, whose origin is on the still water plane. The structures are subjected to a train of regular waves travelling in the positive x-direction.Figure 1
Sketch of the floating structures with a PTO system.The fluid motion is described by the velocity potentialϕ whose spatial gradient is the fluid velocity. The oscillation system is further assumed to be in a steady state. Then, the velocity potential can be written as(1)ϕx,z,t=Re⁡Φx,ze-iωt,where t is the time, i=-1, ω is the angular frequency, Re denotes the real part of a complex, and Φ is a complex spatial velocity potential. Based on the continuity equation, the following Laplace equation can be obtained:(2)∂2Φ∂x2+∂2Φ∂x2=0.In the linear system, the potentialΦ can be further decomposed as(3)Φ=ΦI+ΦD+ΦR,1,where ΦI is the incident potential, ΦD is the diffraction potential, and ΦR,1 is the radiation potential due to the heave motion of the buoy. According to the Airy wave theory, the incident potential has the form as follows:(4)ΦI=-igAωcosh⁡kz+hcosh⁡kheikx,where A is the wave amplitude and k is the wavenumber subjected to the dispersion relation ω2=gktanh⁡(kh). For the diffraction potential, the governing equation is Laplace equation and the boundary conditions can be written as follows:(5)∂ΦD∂z-ω2gΦD=0z=0,x<xl,1orxr,1<x<xl,2orx>xr,2∂ΦD∂z=0z=-h∂ΦD∂z=-∂ΦI∂zz=-dn,xl,n<x<xr,n,n=1,2∂ΦD∂z=-∂ΦI∂z-dn<z<0,x=xr,norx=xl,n,n=1,2ΦDoutgoing:finitevalue,x⟶∞.Here,xl,n and xr,n denote the x coordinate of the left and right side of pontoon n (pontoon 1 refers to the front buoy and pontoon 2 the breakwater).As a preliminary study, the rear breakwater is assumed as a fixed structure, which is constrained by vertical piles. The front buoy works in principle of the heave-type oscillating buoy WEC and the kinetic energy in heave mode of the buoy can be converted into the useful energy by using a PTO system. Thus, we can reasonably assume that the front buoy moves in heave mode and the rear pontoon is fixed. For the radiation problem, the buoy is forced to move in heave mode with amplitudeAR,1 and angular frequency ω. The radiation potential can be expressed as(6)ΦR,1=-iωAR,1φR,1x,z,where AR,1 is the heave amplitude of the buoy. The function of φR,1 also satisfies Laplace equation in the fluid domain. Its boundary conditions can be summarized as(7)∂φR,1∂z-ω2gφR,1=0z=0,x<xl,1orxr,1<x<xl,2orx>xr,2∂φR,1∂z=0z=-h∂φR,1∂z=δm,1z=-dm,xl,m<x<xr,m,m=1,2∂φR,1∂x=0-dm<z<0,x=xr,morx=xl,m,m=1,2φR,1outgoing:finitevalue,x⟶∞.For the solution of the diffraction and radiation potentials, five subdomains I, II, III, IV, and V as indicated in Figure1 are divided. The detailed expressions of the diffraction and radiation potentials can be found in Zheng and Zhang [14], which is not repeated here.Then, the motion equation of the buoy can be written as(8)-ω2M1+μ1-iωλ1+λPTO+K1AR,1=Fz,1,where λPTO is the PTO damping imposed on the front buoy. Unless otherwise noted, λPTO equals the optimal PTO damping (λisolation) of the front buoy in isolation [15, 16]. The added mass μ1 and radiation damping λ1 on the buoy in heave motion can be written as(9)μ1=-ρ∫S1ReφR,1nzdsλ1=-ρω∫S1ImφR,1nzds,where Im denotes the imaginary part of a complex. The horizontal force Fx,n and vertical force Fz,n on pontoon n can be written as(10)Fx,n=-iωρ∫SnΦI+ΦDnxdsFz,n=-iωρ∫SnΦI+ΦDnzds,where Sn is the bottom area of the pontoon n(n=1,2) and nx and nz are the unit normal vector along the negative x-axis and negative z-axis.The CWR as an important efficiency indicator of a WEC is expressed for the front buoy as(11)η=PPincident,where P is the power produced by the buoy as(12)P=12ω2λPTOAR,12.AndPincident is the power of incident waves(13)Pincident=14ρgA2ωk1+2hksinh⁡2hk.The reflection and transmission coefficients (symbolically,KR and KT) can be written as(14)KR=ΦD-iωAR,1φR,1ΦIx=-∞KT=ΦI+ΦD-iωAR,1φR,1ΦIx=+∞.


## 3. Results and Discussions

### 3.1. Verification

In order to validate the analytical solution, the coefficient ofKR2+KT2+η is investigated at different wave conditions. Within the frame of potential-flow theory, the relation of KR2+KT2+η=1 must be satisfied. In this section, we set the geometrical parameters as a1=1 m, a2=6 m, d1=1 m, d2=2.5 m, D=1 m, and h=10 m. From Figure 2, it can be seen that the relation of energy conservation is satisfied, which verifies the accuracy of the present analytical model.Figure 2
Coefficient ofKR2+KT2+η versus the dimensionless wavenumberkh.


### 3.2. Effects of Buoy Breadth

Effects of the buoy on the hydrodynamic performance of the breakwater depend on several parameters, that is, the breadtha1, draft d1, and spacing D. In addition, the hydrodynamic efficiency of the buoy as the wave energy device is affected by those parameters. Therefore, in the following sections, a parametric study is conducted to investigate the effects of these geometrical parameters.The effects of the buoy breadth on the reflection coefficientKR, transmission coefficient KT, and the CWR η can be found in Figure 3. Four cases with a1/h=0.05, 0.1, 0.15, and 0.2 (correspondingly, a1/a2=1/12, 1/6, 1/4, and 1/3) are considered. All the other geometrical parameters are kept constant as a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.1.Figure 3
Variations of reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumber kh for cases with different structure breadths (a1/h=0.05, 0.1, 0.15, and 0.2).
(a)
(b)
(c)Figure3(a) shows the reflection coefficient of the system. The reflection coefficients of breakwaters with and without the buoy are compared. It is found that the existence of the buoy clearly decreases the wave reflection at a wide range of wavenumbers. For the breakwater with a buoy, two troughs can be found on the curve. One is located in the lower frequency region, while the other is in the higher frequency region. In the lower frequency region, a larger buoy breadth generally leads to a weaker wave reflection. However, at the higher frequency region, the trend is opposite. For each KR, both troughs gradually shift to lower frequency regions, as the buoy breadth grows.From Figure3(b), it is found that the buoy breadth does not affect the transmission coefficient evidently. For the CWR in Figure 3(c), the trend of bimodal curve can be described clearly. As the buoy breadth increases, both the peaks and troughs shift to the lower frequency region. The peak positions and trough positions of the CWR curves actually correspond to those positions in Figure 3(a). The existence of the zero value of CWR (in Figure 3(c)) may be due to the fact that the Bragg resonance with strong reflection occurred at certain frequencies, which led to significant reduction of the extraction efficiency [17]. It is similar for the zero values of CWR that occurred in Figures 4(c) and 5(c). Thus, from the viewpoint of engineering application, the occurrence of these should be avoided by the proper structure design based on the dominated sea states at the deployment site. The peaks in the lower and higher frequency regions increase and decrease with a1/h increasing, respectively. It is noted that the maximum of CWR of the oscillating buoy with this configuration is obviously greater than that of the isolated one (i.e., 50%) [15].Figure 4
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different drafts of the front buoy (d1/h=0.1, 0.15, and 0.2).
(a)
(b)
(c)Figure 5
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different spacing (D/h=0.05, 0.1, 0.15, and 0.2).
(a)
(b)
(c)


### 3.3. Effects of Buoy Draft

Figure4 further presents effects of the buoy draft on the integrated system. Three cases are considered, by setting the buoy draft as d1/h=0.1, 0.15, and 0.2 (correspondingly, d1/d2=0.4, 0.6, and 0.8), respectively. The remaining geometrical parameters are kept constant as d2/h=0.25, a1/h=0.1, a2/h=0.6, and D/h=0.1. Similar to Figure 3(a), all curves of reflection coefficients have two troughs. The position of each valley shifts to the lower frequency region, as the buoy draft increases. For the tested cases, as d1/h decreases, the reflection coefficient has more and more significant reduction in the value close to the trough neighborhood. Moreover, regardless of the variation of buoy draft, the reflection coefficient tends to approach that of an isolated breakwater with the increase of wave frequency. Even so, the reflection coefficient for the breakwater with a buoy is generally smaller than that without front buoy for the considered frequency region. Then consider the transmission coefficient in Figure 4(b). It is found that, except for the large draft case (d1/h=0.2), the transmission coefficient of the system is not affected much by the buoy draft. For the CWR found in Figure 4(c), two peaks are observed in the curve. The effect of the draft on η is apparently in the opposite trend to that on KR for kh>2. If we take the frequency region when η>20% as the effective bandwidth, it can be seen that the effective bandwidth becomes narrower with an increasing d1/h.


### 3.4. Effects of Spacing

Effects of the spacing between the buoy and breakwater are further considered in this subsection. The geometrical parameters of two structures are kept constant asa1/h=0.2, a2/h=0.6, d1/h=0.1, and d2/h=0.25. Figure 5 shows variations of the reflection coefficient KR, the transmission coefficient KT, and the CWR η, for different spacing (i.e., D/h=0.05, 0.1, 0.15, and 0.2). The reflection and transmission coefficients corresponding to the breakwater without any buoy are also plotted in the figure for comparison.Firstly, let us consider the reflection coefficients in Figure5(a). When the buoy exists, obvious differences of the curves can be found between kh=2 and kh=8. Two troughs can still be seen in each curve. In the lower frequency region, the troughs value of KR shows an increasing trend as the spacing D/h increases. From Figure 5(b), it is found that the D/h affects the transmission coefficient very slightly. By comparing KR and KT with those of the isolated breakwater, it is found that the value of KR can be reduced by the oscillating buoy, without affecting the value of KT. For the CWR in Figure 5(c), the trend of bimodal curve can also be found in all these four cases. However, both peak values and their corresponding kh vary in different cases. The first peak from the lower frequency region decreases with D/h increasing, but it is opposite for the second peak. Both peaks shift to the lower frequency region as the spacing increases.


### 3.5. Effects of PTO Damping

The optimal damping corresponding to the isolated case is used to calculate CWR of the system in Sections3.2–3.5. The influence of the PTO damping on the hydrodynamics of the system is considered in this subsection. The geometrical parameters of two structures are kept constant as a1/h=0.2, a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.05. Figure 6 shows variations of the reflection coefficient KR, the transmission coefficient KT, and the CWR η, with different PTO damping (i.e., λPTO=0.5λisolation, 0.75λisolation, 1λisolation, 1.25λisolation, and 1.5λisolation).Figure 6
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different PTO damping (λPTO=0.5λisolation, 0.75λisolation, 1λisolation, 1.25λisolation, and 1.5λisolation).
(a)
(b)
(c)Figures6(a), 6(b), and 6(c) show the variations of reflection coefficient KR, transmission coefficient KT, and CWR η versus the dimensionless wavenumber kh, respectively, with different PTO damping. Variation trends of KR versus kh,KT versuskh,and η versuskhfor different PTO damping are similar. For KR versuskh in Figure 6(a), the peak value changed little and the trough value changes significantly, which mainly presents as the first trough value decreasing first and then increasing and the second trough value increasing as the PTO damping increases. However, minor changes can be drawn from the variations of KT versuskh in Figure 6(b). For the CWR, the first peak value increases and then approaches a stable value and the second peak value increases with the increasing of the PTO damping. The wavenumbers corresponding to the trough value of CWR are not sensitive to PTO damping.


### 3.6. Wave Exciting Force on the Breakwater

Since wave exciting force is the key element in design of the marine structures, the horizontal and the vertical wave exciting forces acting on the rear breakwater are of interest. Two scenarios are considered:(1) a case with the front buoy, that is, a1/h=0.2, a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.05 and (2) a case without the front buoy, that is, a1/h=0.0, a2/h=0.6, d1/h=0.0, and d2/h=0.25. Conveniently, we call the former one Case  1 and the latter Case  2. The horizontal force Fx,2 and vertical force Fz,2 are nondimensionalized by ρghA, symbolically, Fx,2/ρghA and Fz,2/ρghA, respectively.Figures7(a) and 7(b) show the variations of Fx,2/ρghA and Fz,2/ρghA versus the dimensionless wavenumber kh, respectively. It can be seen that both Fx,2/ρghA and Fz,2/ρghA of Case  1 are obviously greater than that of Case  2 in the vicinity of kh=5.3. That is to say, the existence of the front buoy may cause the sudden increase of the horizontal force and the vertical force at certain frequencies. This shall be paid attention to while designing such a system from viewpoint of the safety.Figure 7
Variations of the dimensionless horizontal forceFx,2/ρghA and dimensionless vertical force Fz,2/ρghA versus the dimensionless wavenumberkh for Cases  1 and  2.
(a)
(b)


## 3.1. Verification

In order to validate the analytical solution, the coefficient ofKR2+KT2+η is investigated at different wave conditions. Within the frame of potential-flow theory, the relation of KR2+KT2+η=1 must be satisfied. In this section, we set the geometrical parameters as a1=1 m, a2=6 m, d1=1 m, d2=2.5 m, D=1 m, and h=10 m. From Figure 2, it can be seen that the relation of energy conservation is satisfied, which verifies the accuracy of the present analytical model.Figure 2
Coefficient ofKR2+KT2+η versus the dimensionless wavenumberkh.


## 3.2. Effects of Buoy Breadth

Effects of the buoy on the hydrodynamic performance of the breakwater depend on several parameters, that is, the breadtha1, draft d1, and spacing D. In addition, the hydrodynamic efficiency of the buoy as the wave energy device is affected by those parameters. Therefore, in the following sections, a parametric study is conducted to investigate the effects of these geometrical parameters.The effects of the buoy breadth on the reflection coefficientKR, transmission coefficient KT, and the CWR η can be found in Figure 3. Four cases with a1/h=0.05, 0.1, 0.15, and 0.2 (correspondingly, a1/a2=1/12, 1/6, 1/4, and 1/3) are considered. All the other geometrical parameters are kept constant as a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.1.Figure 3
Variations of reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumber kh for cases with different structure breadths (a1/h=0.05, 0.1, 0.15, and 0.2).
(a)
(b)
(c)Figure3(a) shows the reflection coefficient of the system. The reflection coefficients of breakwaters with and without the buoy are compared. It is found that the existence of the buoy clearly decreases the wave reflection at a wide range of wavenumbers. For the breakwater with a buoy, two troughs can be found on the curve. One is located in the lower frequency region, while the other is in the higher frequency region. In the lower frequency region, a larger buoy breadth generally leads to a weaker wave reflection. However, at the higher frequency region, the trend is opposite. For each KR, both troughs gradually shift to lower frequency regions, as the buoy breadth grows.From Figure3(b), it is found that the buoy breadth does not affect the transmission coefficient evidently. For the CWR in Figure 3(c), the trend of bimodal curve can be described clearly. As the buoy breadth increases, both the peaks and troughs shift to the lower frequency region. The peak positions and trough positions of the CWR curves actually correspond to those positions in Figure 3(a). The existence of the zero value of CWR (in Figure 3(c)) may be due to the fact that the Bragg resonance with strong reflection occurred at certain frequencies, which led to significant reduction of the extraction efficiency [17]. It is similar for the zero values of CWR that occurred in Figures 4(c) and 5(c). Thus, from the viewpoint of engineering application, the occurrence of these should be avoided by the proper structure design based on the dominated sea states at the deployment site. The peaks in the lower and higher frequency regions increase and decrease with a1/h increasing, respectively. It is noted that the maximum of CWR of the oscillating buoy with this configuration is obviously greater than that of the isolated one (i.e., 50%) [15].Figure 4
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different drafts of the front buoy (d1/h=0.1, 0.15, and 0.2).
(a)
(b)
(c)Figure 5
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different spacing (D/h=0.05, 0.1, 0.15, and 0.2).
(a)
(b)
(c)


## 3.3. Effects of Buoy Draft

Figure4 further presents effects of the buoy draft on the integrated system. Three cases are considered, by setting the buoy draft as d1/h=0.1, 0.15, and 0.2 (correspondingly, d1/d2=0.4, 0.6, and 0.8), respectively. The remaining geometrical parameters are kept constant as d2/h=0.25, a1/h=0.1, a2/h=0.6, and D/h=0.1. Similar to Figure 3(a), all curves of reflection coefficients have two troughs. The position of each valley shifts to the lower frequency region, as the buoy draft increases. For the tested cases, as d1/h decreases, the reflection coefficient has more and more significant reduction in the value close to the trough neighborhood. Moreover, regardless of the variation of buoy draft, the reflection coefficient tends to approach that of an isolated breakwater with the increase of wave frequency. Even so, the reflection coefficient for the breakwater with a buoy is generally smaller than that without front buoy for the considered frequency region. Then consider the transmission coefficient in Figure 4(b). It is found that, except for the large draft case (d1/h=0.2), the transmission coefficient of the system is not affected much by the buoy draft. For the CWR found in Figure 4(c), two peaks are observed in the curve. The effect of the draft on η is apparently in the opposite trend to that on KR for kh>2. If we take the frequency region when η>20% as the effective bandwidth, it can be seen that the effective bandwidth becomes narrower with an increasing d1/h.


## 3.4. Effects of Spacing

Effects of the spacing between the buoy and breakwater are further considered in this subsection. The geometrical parameters of two structures are kept constant asa1/h=0.2, a2/h=0.6, d1/h=0.1, and d2/h=0.25. Figure 5 shows variations of the reflection coefficient KR, the transmission coefficient KT, and the CWR η, for different spacing (i.e., D/h=0.05, 0.1, 0.15, and 0.2). The reflection and transmission coefficients corresponding to the breakwater without any buoy are also plotted in the figure for comparison.Firstly, let us consider the reflection coefficients in Figure5(a). When the buoy exists, obvious differences of the curves can be found between kh=2 and kh=8. Two troughs can still be seen in each curve. In the lower frequency region, the troughs value of KR shows an increasing trend as the spacing D/h increases. From Figure 5(b), it is found that the D/h affects the transmission coefficient very slightly. By comparing KR and KT with those of the isolated breakwater, it is found that the value of KR can be reduced by the oscillating buoy, without affecting the value of KT. For the CWR in Figure 5(c), the trend of bimodal curve can also be found in all these four cases. However, both peak values and their corresponding kh vary in different cases. The first peak from the lower frequency region decreases with D/h increasing, but it is opposite for the second peak. Both peaks shift to the lower frequency region as the spacing increases.


## 3.5. Effects of PTO Damping

The optimal damping corresponding to the isolated case is used to calculate CWR of the system in Sections3.2–3.5. The influence of the PTO damping on the hydrodynamics of the system is considered in this subsection. The geometrical parameters of two structures are kept constant as a1/h=0.2, a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.05. Figure 6 shows variations of the reflection coefficient KR, the transmission coefficient KT, and the CWR η, with different PTO damping (i.e., λPTO=0.5λisolation, 0.75λisolation, 1λisolation, 1.25λisolation, and 1.5λisolation).Figure 6
Variations of the reflection coefficientKR, transmission coefficient KT, and CWR η versus the dimensionless wavenumberkh for cases with different PTO damping (λPTO=0.5λisolation, 0.75λisolation, 1λisolation, 1.25λisolation, and 1.5λisolation).
(a)
(b)
(c)Figures6(a), 6(b), and 6(c) show the variations of reflection coefficient KR, transmission coefficient KT, and CWR η versus the dimensionless wavenumber kh, respectively, with different PTO damping. Variation trends of KR versus kh,KT versuskh,and η versuskhfor different PTO damping are similar. For KR versuskh in Figure 6(a), the peak value changed little and the trough value changes significantly, which mainly presents as the first trough value decreasing first and then increasing and the second trough value increasing as the PTO damping increases. However, minor changes can be drawn from the variations of KT versuskh in Figure 6(b). For the CWR, the first peak value increases and then approaches a stable value and the second peak value increases with the increasing of the PTO damping. The wavenumbers corresponding to the trough value of CWR are not sensitive to PTO damping.


## 3.6. Wave Exciting Force on the Breakwater

Since wave exciting force is the key element in design of the marine structures, the horizontal and the vertical wave exciting forces acting on the rear breakwater are of interest. Two scenarios are considered:(1) a case with the front buoy, that is, a1/h=0.2, a2/h=0.6, d1/h=0.1, d2/h=0.25, and D/h=0.05 and (2) a case without the front buoy, that is, a1/h=0.0, a2/h=0.6, d1/h=0.0, and d2/h=0.25. Conveniently, we call the former one Case  1 and the latter Case  2. The horizontal force Fx,2 and vertical force Fz,2 are nondimensionalized by ρghA, symbolically, Fx,2/ρghA and Fz,2/ρghA, respectively.Figures7(a) and 7(b) show the variations of Fx,2/ρghA and Fz,2/ρghA versus the dimensionless wavenumber kh, respectively. It can be seen that both Fx,2/ρghA and Fz,2/ρghA of Case  1 are obviously greater than that of Case  2 in the vicinity of kh=5.3. That is to say, the existence of the front buoy may cause the sudden increase of the horizontal force and the vertical force at certain frequencies. This shall be paid attention to while designing such a system from viewpoint of the safety.Figure 7
Variations of the dimensionless horizontal forceFx,2/ρghA and dimensionless vertical force Fz,2/ρghA versus the dimensionless wavenumberkh for Cases  1 and  2.
(a)
(b)


## 4. Conclusions

Hydrodynamic performance of an oscillating buoy WEC installed in front of an existing fixed box-type breakwater was studied systematically with respect to the geometrical and PTO parameters, based on potential-flow theory. The present study reveals that the maximum efficiency of the energy extraction can be 80% or even higher. Compared with the isolated box-type breakwater, the reflection coefficient was reduced effectively by the integration of such an oscillating buoy WEC. However, the transmission coefficients were changed less for the same cases. The possibility of effectively capturing the wave energy by using the oscillating buoy WEC integrated into breakwaters was shown.It should be noted that even though the wave reflection can be greatly reduced at a broad frequency range, there still exist certain frequencies at which the reflection coefficient is still large and the corresponding CWR is close to zero. The obvious increase of the wave exciting force on the rear breakwater can be found at certain frequencies. In the practical applications, the buoy breadth and draft and the spacing between two structures should be optimized to make the high efficiency region located at the dominant wave frequency range and avoid the frequency range with peak wave loads. In addition, at this preliminary stage of the design, only the linear solution within potential-flow theory is discussed in this study. In the future, physical experiments will be further performed to investigate the viscous and nonlinear effects on the hydrodynamic performance of such WEC system.



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