### 1. Introduction

### 2. Mathematical Formulation

*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*and oscillating pressure

_{atm}*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*

^{−}

*} under the assumption of a linear potential theory. The velocity potential*

^{iωt}*ϕ*(

*r*,

*θ*,

*z*), which is a complex function, can be expressed as the sum of the scattering potential (

*ϕ*

_{1}), i.e., the sum of incident potential (

*ϕ*

_{0}) and diffracted potential (

*ϕ*), and the radiated potential (

_{D}*ϕ*

_{2}) due to the heave motion of a floating OWC, and the radiated potential (

*ϕ*

_{3}) 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*

^{−}

*}) of a floating OWC, and*

^{iωt}*p*is the oscillating pressure (

_{c}*P*(

_{c}*t*)=

*Re*{

*p*

_{c}e^{−}

*}) in the chamber. The velocity potential*

^{iωt}*ϕ*, (

_{j}*j*= 1,2,3) in Eq. (1) satisfies the boundary value problem shown below (Eq. (2)).

##### (2)

*k*

_{1}is the wave number, and

*n*is a unit normal vector.

*j*= 1) and radiation (

*j*= 2,3) given above. The hydrodynamic force

*F*(

_{j}*t*)=

*Re*{

*f*

_{j}e^{−}

*} in the vertical direction on a floating OWC and the flow rate (*

^{iωt}*Q*(

_{j}*t*)=

*Re*{

*q*

_{j}e^{−}

*}) due to surface oscillation of interior fluid are expressed as follows. where,*

^{iωt}*S*is the bottom surface of a floating OWC, and

_{B}*S*(=

_{o}*πb*

^{2}) is the free-surface area inside the air chamber. Substituting Eq. (1) in Eq. (3)

*f*

_{1},

*f*

_{2}and

*f*

_{3}can be obtained as follows. where,

*f*

_{1}is the wave exciting force, while

*f*

_{2}and

*f*

_{3}are the hydrodynamic vertical forces on an OWC device by heaving motion of a floating OWC and oscillating pressure in the air chamber, respectively.

*q*

_{1}is the flow rate inside the air chamber due to an incident wave with a unit amplitude, while

*q*

_{2}and

*q*

_{3}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.

### 2.1 Diffraction Problem

*r*≥

*a*,−

*h*≤

*z*≤0; region (II) is defined as

*b*≤

*r*≤

*a*,−

*h*≤

*z*≤−

*d*, and region (III) is defined as 0≤

*r*≤

*b*,−

*h*≤

*z*≤0. If the method of separation of variables is applied to the scattered potential (

*ϕ*

_{1}) using the eigenfunction (cos

*lθ*) in the

*θ*-axis direction, the following equation is obtained.

*l*= 0 in Eq. (7),

*β*= 1, and when

_{l}*l*≥ 1,

*β*= 2(

_{l}*i*)

*.*

^{l}*n*= 0 represents the component of propagating waves, while

*n*≥ 1 means the component of evanescent waves that are only present around objects.

*J*and

_{l}*K*are the Bessel function of the first kind and the modified Bessel function of the second kind, respectively.

_{l}*k*

_{10}=−

*ik*

_{1},

*k*

_{1}

*,*

_{n}*n*= 1,2, …) satisfy the linear dispersion relation (

*k*

_{1}

*tan*

_{n}*k*

_{1}

*=−*

_{n}h*ω*

^{2}/

*g*) where the eigenfunction

*f*(

_{n}*z*) is given by

*f*(

_{n}*z*) defined in Eq. (8) satisfies the orthogonality. where,

*δ*is the Kronecker-Delta function having a value of 1 when

_{mn}*n*=

*m*and a value 0 when

*n*≠

*m*.

*ε*is the Neumann symbol having a value of 1 when

_{n}*n*= 0 and a value 2 when

*n*≥ 1. The eigenvalue in region (II) is

*λ*=

_{n}*nπ*/(

*h*−

*d*), (

*n*= 0,1,2..), and

*R*(

_{ln}*r*),

*R̃*(

_{ln}*r*) has the following form.

##### (11)

*I*is the modified Bessel function of the first kind.

_{l}*A*

_{1}

*,*

_{ln}*B*

_{1}

*,*

_{ln}*B̃*

_{1}

*,*

_{ln}*C*

_{1}

*in Eqs. (7), (10), and (12) are calculated by imposing the continuity of the velocity potential and radial velocity at the interface (*

_{ln}*r*=

*a*,

*b*). The following equation (Eq. (13)) can be obtained by applying the continuity of the velocity potential at

*r*=

*a*. In the process of derivation, the orthogonality of eigenfunction (cos

*λ*(

_{n}*z*+

*h*),

*n*= 0,1,…) in region (II) is applied. where

*r*=

*b*are the same.

*r*=

*a*yields where

*f*(

_{m}*z*),(

*m*= 0,1,2, …), integrating

*z*from −

*h*to 0 and applying the orthogonality of eigenfunction

*f*(

_{n}*z*) given in Eq. (9).

*r*=

*b*are the same. where

*A*

_{1}

*,*

_{ln}*C*

_{1}

*from Eqs. (13), (14), (16), and (17), the algebraic equations of unknowns*

_{ln}*B*

_{1}

*,*

_{ln}*B̃*

_{1}

*can be obtained. The number of eigenfunctions in the (*

_{ln}*z*,

*θ*) direction is set to be finite (

*N*,

*L*). where,

*B*

_{1}

*,*

_{ln}*B̃*

_{1}

*,(*

_{ln}*n*= 0,1,2,.,

*N*,

*l*= 0,1,2,..,

*L*) are determined by solving the algebraic equations given in Eq. (18), and the remaining unknowns

*A*

_{1}

*,*

_{ln}*C*

_{1}

*are determined by substituting them in Eqs. (16) and (17).*

_{ln}*f*

_{1}) in Eq. (4) can be calculating by integrating the scattered potential with respect to the bottom surface of a floating OWC.

### 2.2 Radiation Problem

*j*= 2,3) due to the heaving motion of a floating OWC and oscillating pressure in the air chamber, the fluid region is divided into regions (I), (II), and (III), and the radiated potential in each region is expressed by the series of eigenfunctions, as shown below.

*ψ*(

_{j}*r*,

*z*),

*j*= 2,3 satisfying the body boundary condition

*B*,

_{jn}*B̃*,

_{jn}*j*= 2,3 can be derived from the matching condition where velocity potential and radial velocity are the same at

*r*=

*a*,

*b*. where

*B*,

_{jn}*B̃*,

_{jn}*j*= 2,3 determined by solving the algebraic equation (23), unknowns

*A*,

_{jm}*C*,

_{jm}*j*= 2,3 can be calculated as follows.

##### (26)

*a*

_{33},

*b*

_{33}are called the added mass and radiation damping coefficient.

*B*is radiation conductance, and

*C*is radiation admittance.

### 2.3 Oscillating Air Pressure in the Chamber

*ρ*is the density of air,

_{a}*γ*(= 1.4) is the specific heat for adiabatic situation,

*P*is atmospheric pressure, and

_{atm}*V*(=

_{o}*πb*

^{2}

*H*) is the volume of the air chamber. The flow rate

*Q*(=

_{t}*C*) exiting a turbine is assumed to be linearly proportional to the oscillating pressure in the air chamber.

_{t}P_{c}*C*(=

_{t}*K̂D/N̂ρ*) is the function of the damping coefficient (

_{a}*K̂*) that is dependent on the shape of a turbine, turbine diameter (

*D*), and rotational velocity (

*N̂*) of a turbine.

*m*(=

*ρπ*(

*a*

^{2}−

*b*

^{2})

*d*) is the mass of a floating OWC device.

*f*(=

_{c}*S*) is the force acting on the ceiling of the air chamber by oscillating air pressure, while

_{o}p_{c}*S*

_{1}(=

*π*(

*a*

^{2}−

*b*

^{2})) is the waterplane area of a floating OWC device.

*f*is the damping force due to viscosity; the viscous drag used in the present study is given by the following equation.

_{v}*C*and

_{d}*u*(=

_{r}*u*−

*Av*) are the drag coefficient and the relative heave velocity of a floating OWC device with respect to an incident wave, respectively.

_{z}*v*is the vertical water particle velocity of an incident wave having a unit amplitude. The vertical particle velocity at the bottom surface of a floating OWC, where viscous drag acts predominantly, is used.

_{z}*ω*,2

*ω*,3

*ω*,...) are generated for input frequency

*ω*. Therefore, the velocity potential is expressed as a Fourier series represented in multiples of

*ω*; only the first term is taken while the other terms are assumed to be small enough to be disregarded. The Fourier coefficient of the first term is 8/3

*π*. Eq. (32) can be rewritten as follows through an equivalent linearization. where,

*f*,

_{v}*f*,

_{c}*f*and hydrodynamic forces

_{s}*f*=

*Af*

_{1}+

*uf*

_{2}+

*p*

_{c}f_{3}are substituted in Eq. (31), the equation for the heave motion of a floating OWC device in the frequency domain is as follows:

*u*) and oscillating air pressure (

*p*).

_{c}### 2.4 Extraction Power

*A*is as follows. where

*C*is the group velocity.

_{g}*p*) determined by solving the coupled equation (36) is substituted into Eq. (38) to obtain the time-averaged extracted power.

_{c}*C*)

_{t}*, which gives the maximum time-averaged extraction power of Eq. (38), the equation*

_{opt}*dP̄*/

*dC*= 0 is used. The optimal turbine constant (

_{t}*C*)

_{t}*calculated thereby is as follows.*

_{opt}*u*) of a floating OWC device and oscillating pressure (

*p*) in the air chamber is ignored in Eq. (36), the heave motion velocity and oscillating pressure in the chamber are as follows.

_{c}### 3. Results and Discussions

*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*′ =

*a*

_{33}/

*ρπa*

^{3}) and radiation damping coefficient (

*ν*′ =

*b*

_{33}/

*ρπωa*

^{3}) 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.

*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*) 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}*d*) of a floating OWC, where the natural frequency

*a*

_{33}/

*m*) and radiation damping coefficient (

*b*

_{33}/

*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).

*q*

_{1}) 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 (

*q*

_{3}) 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.

*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.

### 4. Conclusions

The oscillating chamber pressure, which significantly affects the extracted power of a floating OWC device, is considerably influenced by the heave motion of an OWC. Therefore, an equation of motion in which the heave motion of an OWC and oscillating air pressure in the chamber are coupled needs to be solved to accurately evaluate the performance of a floating OWC device. In the present study, a nonlinear viscous drag is considered through an equivalent linearization technique.

When an enclosed fluid region is present inside an OWC, the resonance of internal fluid is generated, and a corresponding double peaks are observed, unlike the motion characteristics of a general floating body exhibiting a single resonance. The heave motion response at the internal fluid resonance frequency is relatively larger than that at the heave natural frequency. The added mass at the internal fluid resonance frequency has a negative value, the corresponding radiation damping coefficient has a peak value. Furthermore, the flow rate of internal fluid due to oscillating pressure in the chamber, called radiation admittance and radiation conductance, shows a similar tendency as added mass and radiation damping coefficient.

The changes in draft and diameter of a floating OWC affect the two different resonance frequencies existing in the hollow cylindrical OWC model. The corresponding peak values and resonance width are changed too with the draft and radius. Therefore, the wave characteristics of the installation site need to be examined in advance to appropriately design the draft and radius for yielding the maximum extraction power.

For a floating OWC device that is identical in shape to a fixed OWC, the power generation efficiency in the long waves is less than that of the fixed OWC. Therefore, the 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 an OWC. Moreover, a latching control technique can be introduced to a floating OWC to maximize extraction efficiency.