The utilization of ocean energy has become an important and urgent issue owing to its abundance and value as clean energy. In particular, the development of new energy sources is inevitably necessary to respond to the carbon emission trading system. However, the commercialization of ocean energy, that has economic feasibility, has not yet been realized because of expensive installation and power transmission costs along with irregularity of the ocean environment. Several efforts have been being taken to resolve these inherent demerits; these efforts include combined power generation, where at least two power sources among wind, wave, and current power are used for a single support structure, direct consumption of generated power at the site, and storing the generated power into the battery without transmission to land.

The technologies related to wave power generation have been developed since the 1980s mainly in Europe and Japan, where wave energy sources are abundant. Several devices have been suggested for wave energy generation, which are near the commercialization stage. Existing wave energy converters (WECs) are categorized as attenuator, point-absorber, and terminator types, based on the position of the WEC with respect to the direction of incident wave propagation. The attenuator type composes of multiple WECs connected parallel to the direction of incident wave propagation; thus electrical energy is generated from the relative motion of converters at the joints by the action of the waves. The Pelamis is a well-known attenuator WEC. The point-absorber type can generate electricity regardless of the direction of incident wave propagation. Particularly, it enhances energy extraction efficiency by amplifying the heave motion of the WEC through resonance with the incident wave with a low energy density. Power buoy and Wavebob are typical point-absorber WEC. Lastly, the terminator type has a WEC positioned in the direction perpendicular to the propagation direction of the incident wave to extract wave energy. Examples of this type include Salter’s duck WEC and dual-functional WEC that has a function of a breakwater.

For converting wave energy into electrical energy, the wave energy is modified through two-step transformations. The first step transformation converts the wave energy into the mechanical energy such as the kinetic and potential energy. These transformation devices are called the movable body, oscillating water column (OWC), and overtopping type (Falcão, 2010). In particular, an OWC device is presently installed and operated in several regions, as the technology has been stabilized. Moreover, the maintenance cost is also fairly low due to a small number of moving machinery parts. The OWC has a enclosed air chamber positioned on top of a supporting structure (fixed, floating) having an open bottom. The free-surface inside the air chamber oscillates up and down due to incident waves, thus inducing airflow due to pressure difference between inside and outside of the chamber; the airflow rotates a turbine located on a passage outside the air chamber; in the final stage, electricity is produced from the generator. The Wells turbine and impulse turbine are used as a second step transformation device that converts the flow energy of air into electrical energy, changing bidirectional airflow into unidirectional turbine’s rotation. The OWC device was first proposed by Masuda (1979); since then, it has been extensively studied by numerous researchers through analytical, numerical, and experimental approaches. The OWC devices are installed in many regions throughout the world, and the Land Installed Marine Powered Energy Transformer (LIMPET) installed in Scotland in 2000 is known as the world’s first wave power generation system that is operated through connection with an existing power grid (Heath et al., 2000). The Korea Research Institute of Ships and Ocean Engineering (KRISO) has recently constructed a real-sea pilot plant that houses fixed type OWC (31.2 × 37.0 m) at water depth 15 m near Chagwido Island, Jeju, and is currently conducting tests to evaluate its performance and efficiency.

To analyze the performance of the fixed and floating OWC device, Evans and Porter (1995, 1997) calculated the wave-energy absorption efficiency by solving the radiation problem due to oscillating pressure in the air chamber and the diffraction problem of a fixed OWC in waves. Mavrakos (1985, 1988) analyzed the diffraction and radiation problems of a hollow cylinder using a matched eigenfunction expansion method (MEEM). Sioris and Memos (1999) performed numerical analysis using the Green’s function method and obtained results that were different from those of Mavrakos (1985, 1988). Cho (2002) used a MEEM to analyze the performance of a fixed cylindrical OWC device. Hong et al. (2004) and Suzuki et al. (2004) numerically analyzed the performance of a floating OWC device using a higher-order boundary element method. Mavrakos and Konispoliatis (2012) used a MEEM to theoretically calculate the extracted power and efficiency of a floating OWC device. Bull (2015) investigated the motion responses and oscillating pressure in the air chambers of a Backward Bent Duct Buoy (BBDB), which belongs to the floating OWC, and an axisymmetric OWC considering the resonance of the internal fluid of an air chamber; they further examined the coupling effect between a floating OWC device and air pressure using a commercially available code WAMIT. Gomes et al. (2012, 2016) carried out systematic researches on the energy extraction performance of an axisymmetric floating OWC device. They replaced the surface of the internal fluid with a thin disk having no weight under the assumption that the size of an air chamber is smaller compared to the wavelength of incident waves and evaluated the extraction power of an OWC device from the relative heave motion between the floating OWC and disk. Regular waves were extended to irregular waves, and the optimization of design parameters was performed using an objective function. Park et al. (2018) evaluated the performance of an axisymmetric floating OWC device using a finite element method (FEM) under a linear potential theory. For considering the damping effects of power take-off, the nonlinear modified kinematic and dynamic free surface boundary conditions proposed by Koo et al. (2010) were linearized to be applied. Furthermore, Luo et al. (2014) applied computational fluid dynamics (CFD) codes for considering viscosity effects. They analyzed the coupling effect of the heave motion of a floating OWC device and oscillating air pressure using a commercially available code FLUENT, which is based on the Navier-Stokes equations.

In this study, the extracted wave power and absorption efficiency of a floating OWC device of a hollow cylinder shape has been extensively investigated in the frequency domain by extending the analytical tool developed by Bae and Cho (2013). Boundary value problems for the diffraction and radiation (heave motion of a floating OWC device, oscillating pressure in the air chamber) were formulated and analytically solved using a MEEM to obtain the hydrodynamic force as well as the flow rate of the internal fluid of the air chamber. Subsequently, the equation of the heave motion of a floating OWC coupled with oscillating pressure in the air chamber was formulated and solved; finally, the extracted wave power and capture width of the OWC device was calculated. Particularly, the piston-mode resonance of the internal fluid in the air chamber was intensively examined. Using the developed predictive tool, the effects of the shape (radius, draft, thickness, etc.) of a floating OWC device and turbine characteristics on the hydrodynamic loads, flow rate, oscillating air pressure, heave motion response, and extraction efficiency were clarified.

A floating OWC device consisting of a hollow cylindrical body with an inner radius b, outer radius a, draft d, and an air chamber with a height H is selected as a calculation model as shown in Fig. 1. The cylindrical coordinate system is introduced, in which the origin is set on the free surface, and the direction of the z-axis is set vertically upward. Air pressure inside the air chamber is expressed as the sum of atmospheric pressure P_atm and oscillating pressure P_c(t) due to waves. The oscillating pressure is assumed to be constant inside the air chamber in space. When the incident wave exhibits a harmonic motion with a frequency ω, the velocity potential can be written by Φ(r,θ,z,t) = Re{ϕ(r,θ,z)e^−iωt} under the assumption of a linear potential theory. The velocity potential ϕ(r,θ,z), which is a complex function, can be expressed as the sum of the scattering potential (ϕ₁ ), i.e., the sum of incident potential (ϕ₀) and diffracted potential (ϕ_D ), and the radiated potential (ϕ₂ ) due to the heave motion of a floating OWC, and the radiated potential (ϕ₃) due to oscillating air pressure in the chamber, as shown in Eq. (1). All motion modes are constrained, and only the heave motion is taken into consideration. Since the calculation model is axisymmetrical, the radiated potential is not associated with the θ-axis, unlike the scattered potential that is a function of (r,θ,z). where A is the amplitude of incident wave, u is the heave motion velocity (U (t)=Re{ue^{− iωt}}) of a floating OWC, and p_c is the oscillating pressure (P_c(t)=Re{p_ce^{− iωt}}) in the chamber. The velocity potential ϕ_j, (j = 1,2,3) in Eq. (1) satisfies the boundary value problem shown below (Eq. (2)). where, k₁ is the wave number, and n is a unit normal vector.

The velocity potential is determined by solving the boundary value problem of the scattering (j = 1) and radiation (j = 2,3) given above. The hydrodynamic force F_j(t)=Re{f_je^{− iωt}} in the vertical direction on a floating OWC and the flow rate (Q_j(t)=Re{q_je^{− iωt}}) due to surface oscillation of interior fluid are expressed as follows. where, S_B is the bottom surface of a floating OWC, and S_o (= πb² ) is the free-surface area inside the air chamber. Substituting Eq. (1) in Eq. (3)f₁, f₂ and f₃ can be obtained as follows. where, f₁ is the wave exciting force, while f₂ and f₃ are the hydrodynamic vertical forces on an OWC device by heaving motion of a floating OWC and oscillating pressure in the air chamber, respectively.

If Eq. (1) is substituted in Eq. (3) in the same manner, the flow rate inside the air chamber can be calculated as follows. where q₁ is the flow rate inside the air chamber due to an incident wave with a unit amplitude, while q₂ and q₃ are the flow rates due to the relative heave motion of a floating OWC and oscillating pressure with a unit pressure inside the chamber, respectively. The flow rate in Eq. (5) causes oscillating pressure inside the air chamber; thus, the turbine operates with the pressure difference from outside.

For verifying the validity of the present solutions obtained using a MEEM, the calculation results are compared with those of Chau and Yeung (2010) as shown in Fig. 2. A hollow cylindrical OWC device has the dimensions of h/a = 1.0, d/a =0.25, b/a = 0.5. The solid line represents the present solutions using a MEEM, while the circle line represents the results obtained by Chau and Yeung (2010). The non-dimensional added mass (μ′ = a₃₃/ρπa³) and radiation damping coefficient (ν′ = b₃₃/ρπωa³) of a floating OWC device agree well with each other. Here, the number of eigenfunctions (N,L) in z and θ directions is 50 and 10, respectively. The same number of eigenfunctions is used in subsequent calculations.

Fig. 3 shows the non-dimensional wave exciting forces with respect to the changes in the draft (d = 6, 8, and 10 m). Here, the x-axis represents the frequency (ω) of the incident wave. The outer radius (a) and inner radius (b) of a floating OWC are 6 m and 4 m, respectively, and the water depth is 30 m. The drag coefficient (C_d) due to viscosity is 0.7. A peak value is observed at a specific frequency within the calculation region, and the peak frequency moves towards the low-frequency region as the OWC’s draft becomes deeper. The presence of these peak values can be explained based on the resonance. Two different types of natural frequency are present in a hollow cylindrical OWC. The first type is the natural frequency of a floating body’s heave motion. The natural frequency of heave motion varies according to the draft (d) of a floating OWC, where the natural frequency ω1(=g/(d+a33/m)) corresponding to three different drafts of 6, 8, and 10 m is 1.19 rad/s, 1.05 rad/s, and 0.95 rad/s, respectively. Specifically, the natural frequency of heave motion decreases as the draft increases. Another type is a piston-mode natural frequency that exists within the enclosed fluid region inside an air chamber. The surface motion at the piston mode oscillates vertically in unison without changes in the form. Fukuda (1977) proposed a simplified equation ω2=g/(d+0.41So) for natural frequency at the piston mode. When the simplified equation of Fukuda (1977) is applied to the present calculation model, the natural frequency of the piston mode is 1.05 rad/s, 0.95 rad/s, and 0.87 rad/s, corresponding to drafts. Thus, the resonance frequency, where peak values are observed in Fig. 3, is closely related to the piston mode resonance.

Fig. 4 shows the non-dimensional added mass (a₃₃/m) and radiation damping coefficient (b₃₃/mω) due to the heaving motion of a floating OWC under the same calculation conditions as those used for Fig 3. The radiation damping coefficient has a peak value when the added mass changes significantly, from a positive to a negative value, at the resonance frequency at the piston mode. This unique phenomenon occurs when a fluid region inducing resonance is present inside an object in motion. This phenomenon is observed in the present model since the fluid region inducing resonance exists within the hollow OWC in heave motion. The added mass and radiation damping coefficient having unique characteristics influence the motion response of a floating OWC device. Only the resonance at the piston mode is displayed since the internal fluid region is smaller than the wavelength of incident wave; however, not only piston mode, but also sloshing mode resonance needs to be considered if the internal fluid region is not smaller than the wavelength of incident wave (Molin, 2001).

Fig. 5 shows the flow rate inside the air chamber due to the incident wave and oscillating pressure in the chamber. First, the flow rate (q₁ ) due to an incident wave when an OWC is fixed has a peak value at the natural frequency of piston mode, similar to a wave exciting force. Meanwhile, the flow rate (q₃) by oscillating pressure in the air chamber has real part (B) and imaginary (C) parts. The curve shapes are similar to those of the added mass and radiation damping coefficient shown in Fig. 4. As the draft decreases, the peak values of B and C also decrease, while the resonance width increases.

Fig. 6 shows the optimal turbine constant that results in maximum power as a function of frequency using Eq. (40). The optimal turbine constant has peak values at specific frequencies. Notably, the peak values of a turbine constant occur at the natural frequency of heave motion. Fig. 6(b) shows the response amplitude operator (RAO) of heave motion of a floating OWC when the optimal turbine constant is applied. The RAO curve of heave motion has two peaks, at the natural frequency of heave motion and at the piston mode natural frequency of the internal fluid. A relatively larger heave motion occurs at the natural frequency of the internal fluid. Fig. 6(c) shows the oscillating pressure inside the air chamber with respect to the frequency of an incident wave when the optimal turbine constant is applied. As predicted, the oscillating pressure has a peak value at the internal fluid resonance frequency, and no significant variation is observed at the natural frequency of heave motion. The extracted power shown in Fig. 6(d) is determined by the turbine constant and the oscillating pressure in the air chamber, as shown in Eq. (38). The frequencies that produce the maximum values of oscillating pressure and turbine constant are different; therefore, two peak values are also observed in the extracted power curve, similar to the heave RAO curve. Fig. 6(e) shows the capture width, determined by dividing the extracted power by the power of incident wave per unit width. As expected, two peaks are present, where the peak value is relatively larger at the internal fluid resonance frequency, similar to the heave RAO and extracted power curves. Moreover, with the increase in the draft of a floating OWC, the peak value at the internal fluid resonance frequency increases while the resonance width decreases; this is because the energy inside the internal fluid region cannot be escaped to the outside. Therefore, the draft of a floating OWC should be designed appropriately for maximizing energy absorption by identifying the wave characteristics of the installation site in advance.

Fig. 7 shows the heave RAO and capture width of a floating OWC with respect to the inner radius (b) of 3 m, 4 m, and 5 m, which is another important design parameter. The draft (d) and outer radius (a) of a floating OWC are fixed at 8 m and 6 m, respectively. The drag coefficient of a floating OWC is 0.7, and the optimal turbine coefficient is used. The heave natural frequency of a floating OWC within the calculated frequency region is 1.03 rad/s, 1.05 rad/s, and 1.07 rad/s as the inner radius increases; meanwhile, the resonance frequency of internal fluid in the piston mode is 0.98 rad/s, 0.95 rad/s, and 0.92 rad/s, thus being unaffected greatly by the inner radius of a floating OWC. However, the peak value of heave motion amplitude at the internal fluid resonance increases as the inner radius increases. In the maximum capture width curve obtained by applying optimal turbine constant, the capture width has one peak at the internal fluid resonance frequency when the inner radius is the largest at 5 m; the peak value decreases when the inner radius is the smallest at 3 m, and it increases again at the heave natural frequency, thus yielding two peak values. Consequently, the peak value decreases with the decrease of inner radius, but the frequency range for wave energy extraction increases. Similar to the draft, the inner radius of a floating OWC also needs to be designed appropriately for the wave characteristics of the installation site.

Fig. 8 shows a comparison of the coupled results between the oscillating pressure and the heave motion of a floating OWC with the uncoupled results obtained using Eq. (41). The heave RAO illustrated in Fig. 8(a) shows that the coupled heave RAO has a decreased peak value at the heave resonance frequency due to the influence of oscillating pressure inside the air chamber. The oscillating pressure inside the air chamber illustrated in Fig. 8(b) shows that the results of oscillating pressure uncoupled with a floating OWC are identical to the results of the fixed type of an OWC. Thus, when the oscillating chamber pressure and heave motion of a floating OWC are coupled, the oscillating pressure is closer to zero due to a small relative heave motion as a floating OWC moves along with the incident long waves; subsequently, a peak value is observed at the internal fluid resonance frequency due to the large displacement of the internal fluid. As predicted from Fig. 8, the power generation of a floating OWC is less efficient than that of a fixed OWC in long waves. Therefore, it is recommended to design the system such that the heave motion of the OWC is minimized. To this end, a damping plate can be installed at the bottom of a floating OWC, or a tension-leg mooring system can be introduced to minimize the heave motion of the OWC. Furthermore, a latching control technique, which is often adopted in a movable-type WEC, can be introduced to a floating OWC to maximize the extraction efficiency.

The following conclusions have been drawn from investigation of the heave RAO, oscillating air pressure, extracted power, and capture width of a floating hollow cylindrical OWC device with respect to changes in draft and radius of an OWC.