### 1. Introduction

### 2. Mathematical Formulation

*V*and

*ϕ*denote the velocity of the fluid particle and velocity potential, respectively. The velocity potential can easily represent a complex fluid flow phenomenon as a scalar value, and when the fluid flow satisfies the continuity equation and is an incompressible flow, the governing equation of the fluid domain can be represented using the Laplace equation:

^{nd}identity, as expressed in Eq. (3). where

*Ω*denotes the fluid domain in which the calculation is performed. The solid angle

*α*at the boundary is 0.5, and the two-dimensional Green’s function is

*G*=− (1/2

_{ij}*π*)ln

*R*

_{1}. Here,

*R*

_{1}represents the distance between the source point and the field point at the boundary of the computation domain.

*δ*/

*δt*). In addition, under the assumption that the node points located on the free surface move in the same manner as the water particles (Material node approach,

*ν⃗*= ∇

*ϕ*), they can be defined as expressed in Eqs. (4) and (5). where

*η*,

*g*,

*ρ*, and

*P*denote the free surface elevation, gravity, fluid density, and pneumatic pressure on the free surface, respectively. Outside the chamber, the pneumatic pressure is zero, based on atmospheric pressure, and when pneumatic pressure is present inside the chamber, its value changes in real time, according to the volume of the OWC chamber and the size of the nozzle (duct).

_{a}*K*and

_{L}*Ṽ*represent the loss coefficient and flow velocity, respectively. This equation can be expressed as Eq. (8), under the assumption that the water particle velocity (

*∂ϕ*/

*∂n*) is linearly proportional to the average vertical velocity of the water column (

*η̇̄*). The viscous damping effect, which can occur when the incident wave enters the chamber, can be represented using the viscous damping coefficient (

*ν*). By applying Eq. (8) to the dynamic free surface boundary condition in the chamber (

*δη*/

*δt*=

*∂ϕ*/

*∂z*), the energy reduction of the incident wave owing to fluid viscosity can be expressed as a viscous damping pressure term (Yang et al., 2019). where

*η̇̄*represent the modified loss coefficient and average velocity of the surface elevation inside the OWC chamber, respectively. Inside the OWC chamber where the duct is installed, the pneumatic pressure is generated by the free surface elevation, and this is represented by assuming a linear relationship between the pneumatic pressure (

*P*) in the OWC chamber and the velocity of the air flowing through the duct (Ning et al., 2015; Yang and Koo, 2020). where

_{ac}*C*denotes a linear pneumatic damping coefficient and

_{dm}*U*(

_{d}*t*)=

*U*

_{0}sin

*ωt*represents a relationship between air flow rate with time. The air flow rate through the duct is determined by the ratios of the chamber and duct sizes. Therefore, the pneumatic pressure inside the chamber is defined as: where

*A*represents the area of the duct,

_{d}*ΔV*/

*Δt*is the rate of volume change inside the chamber, and the free surface boundary condition inside the OWC chamber, considering both the viscous damping effect and the pneumatic pressure effect (extractable by energy) owing to the duct, can be expressed by Eq. (11).

*H*and

_{i}*H*denote the incident and reflected wave heights, respectively, and

_{R}*C*represents the group velocity. The average energy extraction rate (

_{g}*AP*) is a value calculated per OWC chamber area and incident wave period, and it can be observed that the rate is proportional to the square of the air flow velocity in the nozzle (duct). The energy loss (

*EL*) owing to viscosity was also calculated similarly, and it was confirmed to be proportional to the square of the average vertical velocity (

*η̇̄*) of the free surface in the chamber. Therefore, the sum of the reflected wave energy flux (

*E*), average energy extraction (

_{R}C_{g}*AP*), and energy loss (

*EL*) must be equal to the incident wave energy flux (

*E*) (

_{i}C_{g}*E*+

_{R}C_{g}*AP*+

*EL*=

*E*). Using this equation, the validity of this numerical modeling was verified via the numerical analysis results, which indicate that the energy of the entire WEC system is conserved within the computation domain

_{i}C_{g}### 3. Numerical Modeling and Results

*d*and

*B*denote the draft and thickness of the OWC chamber skirt, respectively,

*L*represents the width (gap) of the chamber, and

*h*represents the water depth. In addition,

*B*and

*h*are 0.09 m and 0.5 m, respectively, and these values were fixed in this study, and the slope angle of the skirt was set at 33.7°, which is similar to the angle of domestic sloped breakwaters, and would enable the installation of this model on a sloped breakwater in the future.

### 3.1 Open Chamber

*ν*= 0, 0.5, respectively. This corresponds to

*λ*/

*L*= 13.8 in Fig. 3. The surface elevation is the value of the difference between the wave crest and wave trough divided by 1/2. In this study, a ramp function twice the incident wave period (

*T*) was applied to generate stable incident waves in the fully nonlinear numerical wave tank. In addition, because OWC WEC is located at a distance approximately 4 times the incident wavelength from the wave maker of the numerical wave tank, it can be inferred that the surface elevation in the chamber occurs after approximately 4 periods (

*t*= 4

*T*). Under the above calculation conditions, in the calculation of the pure potential flow without viscous damping, it can be deduced that the elevation of the free surface inside the chamber is approximately 5 times higher than the incident wave, and when viscous damping effect is included, the elevation is approximately 3 times higher. Because the surface elevation is high, as obtained from the results of the nonlinear calculation, the heights of the wave crest and wave trough are different, which triggers the asymmetry between the upper and lower surface elevations.

*ν*= 0.5) was applied to the numerical model, and those obtained by pure potential flow analysis (

*ν*= 0) are compared with the experimental data from Case 1. It can be observed that the results of the fully nonlinear numerical analysis with appropriate viscous damping coefficients agree well with the results of the 2D water tank experiment. In addition, when viscous damping was applied, the wave elevation in the chamber was reduced by approximately 28%. In particular, it was verified that when the incident wavelength is longer than the chamber width (approximately 12–15 times), the energy attenuation due to viscosity increases. In the conditions of this study, it can be observed that the viscous damping effect was the largest when the incident wavelength was approximately 14 times the chamber width.

*ν*= 0.7 in Case 2,

*ν*= 0.8 in Case 3,

*ν*= 0.75 in Case 4, and

*ν*= 0.8 in Case 5 were applied to the numerical model, the results were closest to the experimental results. Therefore, the reliability of the OWC chamber numerical modeling (open chamber condition) was verified by calculating the appropriate viscous damping coefficient according to the chamber skirt draft and chamber width compared to the experimental values.

### 3.2 Partial Open Chamber with Duct

*C*) represents the maximum extractable wave energy that can be extracted from the air turbine installed in the duct, and it is related to the ratio of the chamber and duct sizes. The pneumatic damping coefficient value (

_{dm}*C*= 50) was obtained by comparing the numerical results with the experimental results reported in Park et al. (2018a). The ratio of the chamber cross-sectional area to the duct area was approximately 120. For the viscous damping coefficient, values obtained from a structure of the same size under the open chamber conditions were adopted.

_{dm}*d*= 0.1 m,

*L*= 0.25 m) conditions, with the incident wave period of 1.75 s and wave height of 0.051 m, the incident wave and free surface elevation in the chamber were compared in time series (Fig. 6). Unlike the open chamber condition, the free surface elevation in the chamber was reduced to approximately 1/2, compared to the incident wave height, and to approximately 1/4 compared to the open chamber results, as the wave energy was extracted owing to the presence of the duct. To verify the pneumatic pressure effect, the results were compared with the results of the water tank experiment conducted by Park et al. (2018a), under the same conditions (Fig. 7). In the open chamber (

*C*= 0) case, i.e., when the pneumatic pressure was zero, as the incident wavelength increased, the water surface in the chamber was elevated up to approximately 2.5 times the incident wave height; however, when a duct was installed in the chamber, the surface elevation decreased. This is because the pressure in the air chamber increased, owing to the installed duct; hence, free surface elevation in the chamber was suppressed. In fact, this can be regarded as the same effect when the wave energy was extracted with the air turbine of OWC, which led to the decrease in the energy in the chamber, and reduced the surface elevation. The above calculation is a case in which the air in the chamber is smoothly discharged from the duct without excessive compression, assuming that an air turbine with a large diameter, such as a Wells or an impulse turbine, is mainly used in the OWC. Therefore, in the case in which the viscous damping term has already been considered, the difference between the surface elevation in the open chamber and that in the chamber with the duct installed can be ascertained as the maximum extractable wave energy by the air turbine.

_{dm}*d*= 0.1 m,

*L*= 0.25 m). It can be observed that the sum of the reflected wave energy flux, viscous damping energy, and air energy in the chamber is almost identical to the incident wave energy flux. This indicates that all energy flux components are accurately measured in the OWC WEC system. It can be observed that the energy (E-air) due to the air flow in the chamber increases as the incident wave period increases, which is approximately 20–40% of the incident wave energy. In addition, it can be observed that the reflected wave energy decreases with the increase in the period, which is at least 50% or more of the incident wave energy. In the future, it is necessary to develop a method that minimizes the reflected wave energy, such that more wave energy can be extracted. No significant change exists in the viscous damping energy relative to the change in the incident wave period.

### 3.3 Real Scale Model

*L*= 5 m,

*d*= 2 m), which is enlarged by 20 times compared to the scale model (

*L*= 0.25 m,

*d*= 0.1 m) under Case 1(

*L*/

*d*= 2.5). Accordingly, the changes in the viscous damping coefficient (

*ν*/

*ν*

_{0},

*ν*

_{0}is the viscous coefficient of the scale model (

*d*= 0.1 m) can be illustrated in Fig. 10 for various scale factors (2, 5, 10, 15, 20 times the initial scale model) of Cases 2, 4, and 5, with the same chamber width, and Cases 1–3 with the same draft of the chamber skirt. The obtained results confirmed that the viscous damping coefficient increased with a similar trend, according to the scale in all Cases.

*C*) is related to the volume ratio,

_{dm}*C*= 50 used in the scale model was adopted without alterations. When comparing the maximum extractable energy under the given incident wave conditions, the obtained value was the largest in Case 1. Table 3 presents the maximum extractable energy under the incident wave conditions, as well as the shape of Case 1 for the conditions. Under the conditions of period (

_{dm}*T*= 7.83 s), wavelength (

*λ*= 75.41 m), and wave height (

*H*= 1.02 m), the value of the maximum extractable energy was up to 4.59 kW/m, which is approximately 40% of the incident wave energy.