On-Shore U-OWC Wave Energy Converter: A Hydrodynamic Study of Its Capture Performance Impacted by Air-Compressibility Effects

1. Introduction

Marine renewable energy has increasingly emerged as a cornerstone in the pursuit of sustainable and resilient global energy systems. Among the various ocean-based resources, wave energy stands out for its exceptionally high energy density and inherent predictability, offering clear advantages over other renewables. As reported by Falnes [1,2], typical marine wave energy densities range from approximately 2 to 3 kW/m² — several times greater than those of wind (0.4 – 0.6kW/m²) and solar radiation (0.1 – 0.2kW/m²). Globally, wave energy accounts for nearly 39% of total marine renewable potential, representing a theoretical annual resource of roughly 29,500 TWh/year [3]. These characteristics make wave energy a compelling and strategically significant renewable resource for addressing long-term energy security and climate challenges.

Among various wave energy converter (WEC) technologies, the Oscillating Water Column (OWC) has gained particular prominence due to its structural simplicity and demonstrated reliability in harsh marine conditions. This is because OWC systems convert the oscillatory motion of enclosed water columns into pneumatic power, which subsequently drives the power take-off (PTO), i.e., the air turbine, which is inherently located away from erosive and biohazardous seawater. Furthermore, the on-shore type of OWC has one additional key advantage: its compatibility with existing coastal infrastructure, such as breakwaters [4] and harbor walls, which further enhances its survivability while reducing construction and maintenance costs. Notable full-scale installations of on-shore OWCs include the Sakata Harbor wave power plant in Japan [5], the U-shaped OWC at the Mutriku wave power plant in Spain [6], and the port of Civitavecchia in Italy [7]. Collectively, these implementations underscore the on-shore OWC’s Technology Readiness Level (TRL 8) [8], economic feasibility, and suitability for deployment, thereby positioning it as one of the most extensively researched and practically demonstrated WEC technologies.

The historical development of OWC technology traces back to an early navigational aid patented by Courtney in the 1880s, the whistling buoy [9], which harnesses wave-induced pressure fluctuations to produce acoustic signals. This OWC device also marks the first practical demonstration of ocean energy conversion. In the early 20th century, Bochaux-Praceique constructed a small-scale OWC near Bordeaux, France, capable of producing roughly 1 kW of power for household use [10]. Masuda’s wave-powered navigation buoy exemplified subsequent progress in 1947 [11], which incorporated an air turbine for charging batteries and for autonomous power generation. The modern era of OWC research emerged during the energy crises of the 1960s and 1970s, motivating intensive theoretical, numerical, and experimental studies of OWC hydrodynamics and performance. Major contributions include theoretical analyses (e.g., Evans [12]; Malmo and Reitan [13]), numerical investigations (e.g., Hong et al. [14]; Simonetti et al. [15]; Wang et al. [16]), and experimental validations (e.g., Sarmento [17]; López et al. [18]). A comprehensive overview of OWCs’ evolution is provided by Falcão and Henriques [19]. As a whole, these efforts have significantly advanced the scientific understanding and technological readiness of OWC-based wave energy conversion.

With the advent of advanced computational capabilities, Computational Fluid Dynamics (CFD) has become an indispensable tool for OWC research, enabling robust analyses of complex wave–structure interactions without the prohibitive costs of large-scale experiments. CFD-based investigations have yielded valuable insights into OWC optimization, for example, Bouali and Larbi [20] employed ANSYS Fluent to explore geometric configurations that maximize energy extraction; López et al. [21] used STAR-CCM+ with a RANS–VOF framework to model turbine–chamber coupling; Ciappi et al. [22] validated aerodynamic performance of Wells turbines via hybrid analytical–CFD models. Even though some numerical models other than CFD have been particularly used, e.g., Kim et al. [23]—potential-flow simulations for inclined OWC geometries, Ning et al. [24]—fully nonlinear time-domain boundary element model for wave–structure interaction, Rezanejad et al. [25]—two-dimensional linear wave theory for analyzing OWCs with stepped bottoms, current research trends underscore CFD’s pivotal role in improving modeling accuracy, accelerating design optimization, and bridging gaps between theory and experiment in modern OWC developments.

Recent comparative numerical studies have increasingly focused on geometric configurations. Three primary geometric configurations were commonly employed in the design and study of OWCs: the traditional OWC, L-shaped OWC (L-OWC), and U-shaped OWC (U-OWC). López et al. [26] conducted 2D numerical simulations with OpenFOAM to evaluate and compare these three configurations at a model scale, and to validate their numerical results against experimental data. Their study focused specifically on identifying optimal dimensions for U-OWC and L-OWC systems operating under the wave climates at Vigo, Spain. The L-OWC configuration outperformed both U-OWC and traditional models in energy efficiency; however, the U-OWC was distinguished by its broader operational bandwidth. Based upon this, Lin et al. [27] undertook 2D numerical simulations of incompressible flow by FLOW-3D^® to establish an extensive performance database for a model-scale L-OWC. They subsequently used an optimization process integrated with a machine learning process to systematically yield optimal structural dimensions, achieving further improvements in energy capture efficiency compared to those reported by López et al. [26]. These findings underscore the importance of geometric configuration in OWC performance, highlighting the ongoing need for detailed numerical analyses and comprehensive hydrodynamic studies of OWC geometries.

Despite considerable progress, several challenges persist in the numerical simulation and practical optimization of OWCs. Previous studies by López et al. [26] and Lin et al. [27] identified four primary challenges frequently encountered during numerical investigations of OWCs, namely scaling effects between model-scale and full-scale simulations, dimensional simplifications from 2D to 3D modeling, accurate modeling of air compressibility, and realistic representation of Power Take-Off (PTO) mechanisms. Among these factors, accurately capturing the air compressibility effects within the plenum chamber of OWCs is particularly critical, as they significantly influence the device’s performance at full-scale conditions. Sarmento and Falcão [28], Portillo et al. [29], and Falcão and Henriques [30] characterized the dynamic effects of air compressibility in OWCs by analogy to a spring-like mechanism. Numerical studies of a 3D, full-scale OWC by Elhanafi et al. [31] revealed that failing to account for air compressibility can result in efficiency errors of nearly 12% compared to model-scale data, particularly under resonance and optimized PTO damping. Further numerical investigations by Simonetti et al. [32] using OpenFOAM reinforced these conclusions, indicating that omitting air compressibility could result in overestimations in both the plenum chamber air pressure and volumetric airflow by as much as 15%. Nevertheless, the inaccuracies in capture efficiency remained largely within a 10% margin. Carlo et al. [33] focused on optimizing the hydrodynamic performance and geometric parameters of a traditional U-OWC. Combining 1:30-scale physical wave flume experiments with CFD simulations, this study explored the effects of the U-duct width and the orifice diameter on the device's performance. Furthermore, it derived empirical relationships for calculating the resonance index and hydrodynamic efficiency for engineering design. Wang et al. [34] proposed an innovative U-OWC concept featuring a front wall capable of pitching motion within the U-duct. Based on linear wave theory analysis, the study demonstrated that by configuring an appropriate angular spring stiffness, this dynamic front wall design can significantly expand the frequency bandwidth for high-efficiency wave energy absorption. However, the enhanced efficiency results in significantly larger wave loads acting on the back lip-wall, necessitating a balance between wave power extraction efficiency and structural safety in practical applications. Zheng et al. [35] similarly aimed to modify front wall rigidity by proposing a U-OWC design with a flexible bottom-standing front wall. By developing a theoretical model based on linear potential flow theory, it is found that the flexural rigidity of the flexible wall is a key parameter. Its deflection can induce both the natural mode resonance of the flexible wall and wave near-trapping effects, leading to three distinct peaks in the maximum wave power capture efficiency curve, thereby achieving high-efficiency absorption over a broader range of wave frequencies. Model experiments on an OWC chamber connected to an expanded reservoir performed by López et al. [36] further corroborated these numerical insights, demonstrating that disregarding compressibility effects could lead to significant underestimations or overestimations of power output, depending heavily on specific wave conditions and turbine-induced damping. Additionally, a theoretical correction method proposed by Falcão et al. [37], combining experimental and potential flow analyses, demonstrated that appropriately accounting for air compressibility can notably enhance predicted capture width ratios at specific wave periods. Altogether, these studies underscore the complexity and importance of accurately accounting for air compressibility in numerical simulations to ensure reliable optimization of OWC performance. Henriques et al. [38] conducted wave flume experiments on a breakwater-integrated U-OWC, uniquely incorporating accurate modeling of air compressibility into the scaled model. The experimental results confirmed that the ratio of wave amplitude to lip clearance significantly affects device efficiency; increasing wave amplitude or decreasing lip clearance reduces the pneumatic capture width ratio. This non-linear effect implies that the U-OWC possesses the potential for "passive control of pneumatic power peaks," allowing the system to self-protect during highly energetic wave conditions.

To address these challenges, Nguyen et al. [39] introduced a scaling-rematched methodology that provides a systematic framework for reconciling model-scale and full-scale hydrodynamics. This approach enables the accurate determination of essential coefficients, such as added mass, radiation damping, and excitation forces, while integrating theoretical corrections to account for spring-like air compressibility effects. The method further employs FLOW-3D’s impeller model to simulate PTO damping effects from air turbines, enhancing the fidelity of numerical predictions. Applied initially to an L-OWC, the methodology proved effective in refining performance estimates and quantifying the influence of compressibility. The present paper further applies this methodology to a U-OWC and comprehensively evaluates its capture performance, hydrodynamic behavior, and compressibility effects. This paper is structured as follows. Section 2 briefly describes the scaling-rematched approach and details the numerical simulations employed. Section 3 presents the results and discussions of capture factors, hydrodynamic and gravitational coefficients, and effective coefficients of PTO damping and air compressibility. Lastly, Section 4 concludes the main results.

2. Methodology

2.1. Scaling-Rematched Approach

This study adopts the theoretical approach developed by Nguyen et al. [39] for rematching the scales associated with hydrodynamics and air compressibility. In this framework, the free surface of the water column within an OWC’s plenum chamber, located at a depth h, is modeled as a massless oscillator that undergoes vertical harmonic oscillations in response to an incident plane wave of amplitude at the angular frequency $\omega$. Nguyen et al. [39] proposed a key modeling which serially connects effects of PTO damping and air compressibility caused by the air turbine on the oscillator (OWC’s free surface) that yields “effective” coefficients of air compressibility and PTO damping, respectively ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$, as

$$\left\{\begin{array}{l}{\widehat{C}}_{air}=\left({\displaystyle \frac{{\omega }^2{\nu }_{PTO}^2}{{\omega }^2{\nu }_{PTO}^2+a_S^4{C}_{air}^2}}\right)C_{air}\\ {\widehat{\nu }}_{PTO}=\left({\displaystyle \frac{a_S^2{C}_{air}^2}{{\omega }^2{\nu }_{PTO}^2+a_S^4{C}_{air}^2}}\right){\nu }_{PTO}\end{array}\right.,$$

(1)

where $a_S$ represents the cross-sectional area ratio between the PTO and the chamber's free surface, $C_{air}$ is the coefficient capturing the spring-like effect of air compressibility inside the OWC, and ${\nu }_{PTO}$ is the damping coefficient attributed to the PTO (air turbine). The key element in this modelling, $C_{air}$, can be further derived as ([28])

$$C_{air}={\displaystyle \frac{k_1{p}_{atm}{S}^2}{{\mathrm{V}}_o}},$$

(2)

where $k_1$ denotes the exponent in the polytropic equation of air, $p_{atm}$ the atmospheric pressure of air, S the area of the free surface in the OWC’s plenum chamber (the chamber’s cross-sectional area), and ${\mathrm{V}}_o$ the chamber’s volume above SWL.

As noted by Nguyen et al. [39], acquiring authentic, full-scale experimental data regarding air compressibility in OWCs remains a significant challenge. Furthermore, to the authors’ best knowledge, no full-scale OWC experimental data are available. Nevertheless, we can perform certain inductive verifications of the modelling in Eq. (1) by demonstrating its capability to yield the expected limiting behaviors of ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$ as follows.

• As ${\nu }_{PTO}$ approaches zero (i.e., the air chamber is fully open to the atmosphere without PTO/air turbine present), the formulation rightly forces both ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$ to vanish. Such a result validates that no air volume fluctuates and no power is captured without PTO.
• As $C_{air}$ approaches zero, which represents a vacuum condition within the plenum chamber, the model dictates that both ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$ approach zero. This outcome physically signifies that without an air medium, neither air compressibility nor the damping effects of the PTO system can exist.
• As ${\nu }_{PTO}$ approaches infinity, which represents a completely blocked PTO where the plenum chamber acts as a closed space, the model predicts that ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$ approach $C_{air}$ and zero, respectively. This outcome indicates that only air compression and expansion occur, resulting in zero power extraction.
• As $C_{air}$ approaches infinity, the air within the plenum chamber is treated as incompressible. This is a standard assumption in model-scale studies, where ${\mathrm{V}}_o$ in Eq. (2) is typically small. The modeling yields ${\widehat{C}}_{air}$ and ${\widehat{\nu }}_{PTO}$ rightly as

$$\left\{\begin{array}{l}{\widehat{C}}_{air}\to {\widehat{C}}_{air,\infty }=0\\ {\widehat{\nu }}_{PTO}\to {\widehat{\nu }}_{PTO,\infty }=a_S^{-2}{\nu }_{PTO}\end{array}\right.,$$

(3)

where the subscript “$\infty$” denotes that $C_{air}\to \infty$. This result confirms that, by definition, incompressible flows exhibit no compressibility effects.

Nguyen et al. [39] further show that the capture factor $C_F$ of an OWC can be derived as

$$C_F=\left({\displaystyle \frac{{\left|F_E\right|}^2}{4\rho g{\left|A_i\right|}^2{V}_{g}B{\nu }_f}}\right)\left[{\displaystyle \frac{4\left({\widehat{\nu }}_{PTO}\cdot {\nu }_f\right)}{{\omega }^{-2}{\left({\widehat{C}}_{air}+\rho gS-{\omega }^2{\mu }_A\right)}^2+{\left({\widehat{\nu }}_{PTO}+{\nu }_f\right)}^2}}\right],$$

(4)

where $F_E$ denotes the complex amplitude of the wave exciting force, ${\nu }_f$ the damping coefficient resulting from the fluid effects of wave radiation and viscosity, ${\mu }_A$ the added mass induced by the water column’s oscillation, B the OWC’s width, $V_g$ the group velocity of the incident wave, $\rho$ the water density, and g the gravitational acceleration. This suggests that Eq. (4) can be expressed in a dimensionless format as

$$C_F=\left[{\displaystyle \frac{{\left({\left|F_E\right|}^{*}\right)}^2}{{\nu }_f^{*}}}\right]\left[{\displaystyle \frac{4{\widehat{\nu }}_{PTO}^{*}\cdot {\nu }_f^{*}}{{\left({\widehat{C}}_{air}^{*}+g^{*}-{\mu }_A^{*}\right)}^2+{\left({\widehat{\nu }}_{PTO}^{*}+{\nu }_f^{*}\right)}^2}}\right],$$

(5)

where

$$\left({\left|F_E\right|}^{*},{\mu }_A^{*},{\nu }_f^{*},g^{*}\right)\equiv \left({\displaystyle \frac{\left|F_E\right|}{2\rho \left|A_i\right|B\sqrt{gh}{V}_g}},{\displaystyle \frac{{\mu }_A}{\rho \left(Bh/\omega \right)V_g}},{\displaystyle \frac{{\nu }_f}{\rho \left(Bh\right)V_g}},{\displaystyle \frac{g}{\omega \left(Bh/S\right)V_g}}\right),$$

(6)

and

$$\left(C_{air}^{*},{\widehat{\nu }}_{PTO,\infty }^{*},{\widehat{C}}_{air}^{*},{\widehat{\nu }}_{PTO}^{*}\right)\equiv \left({\displaystyle \frac{C_{air}}{\rho \omega \left(Bh\right)V_g}},{\displaystyle \frac{a_S^{-2}{\nu }_{PTO}}{\rho \left(Bh\right)V_g}},{\displaystyle \frac{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2{C}_{air}^{*}}{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2+{\left(C_{air}^{*}\right)}^2}},{\displaystyle \frac{{\left(C_{air}^{*}\right)}^2{\widehat{\nu }}_{PTO,\infty }^{*}}{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2+{\left(C_{air}^{*}\right)}^2}}\right)$$

(7)

It can be seen that the second bracket on the right-hand side of Eq. (5) attains its maximum of 1 when ${\widehat{C}}_{air}^{*}={\mu }_A^{*}-g^{*}$​ that achieves reactance cancellation and ${\widehat{\nu }}_{PTO}^{*}={\nu }_f^{*}$ that satisfies resistance matching.

To initiate the scaling-rematched approach, Eq. (5) is reformulated for the model scale (denoted by the subscript “ms”) when ${\widehat{C}}_{air}^{*}\to 0$, as suggested by Eq. (3). That is,

$$C_{F,ms}={\displaystyle \frac{4{\left({\left|F_E\right|}_{ms}^{*}\right)}^2\cdot \left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)}{{\left(g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)}^2+{\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}+{\nu }_{f,ms}^{*}\right)}^2}}$$

(8)

Once Froude similarity is established, the square of the wave frequency (${\omega }^2$) scales inversely with the geometric dimension of the OWC. Consequently, the hydrodynamic and gravitational coefficients on the right-hand side of Eq. (8) are expected to conform to the model law, i.e.,

$${\left({\left|F_E\right|}^{*},{\mu }_A^{*},{\nu }_f^{*},g^{*}\right)}_{ps}={\left({\left|F_E\right|}^{*},{\mu }_A^{*},{\nu }_f^{*},g^{*}\right)}_{ms},$$

(9)

where the subscription “ps” denotes the prototype scale.

Next, by applying regression and curve-fitting techniques to the database $\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*},C_{F,ms}\right)$ obtained from either numerical simulations or model-scale experiments, the hydrodynamic parameters ${\left({\left|F_E\right|}^{*},{\mu }_A^{*},{\nu }_f^{*},g^{*}\right)}_{ms}$ can be evaluated to represent ${\left({\left|F_E\right|}^{*},{\mu }_A^{*},{\nu }_f^{*},g^{*}\right)}_{ps}$.

In order to rematch the scaling for the air compressibility, $C_{air,ps}$ is evaluated using Eq. (2) as

$$C_{air,ps}={\displaystyle \frac{\gamma p_{atm}{S}_{ps}^2}{{\mathrm{V}}_{o,ps}}},$$

(10)

where $\gamma =k_{1,ps}=1.4$ (isentropic process) as proposed by Falcão et al. [24]. By letting ${\widehat{\nu }}_{PTO,\infty ,ps}^{*}={\widehat{\nu }}_{PTO,\infty ,ms}^{*}={\widehat{\nu }}_{PTO,\infty }^{*}$ and using Eq. (7), the effective coefficients of air compressibility and PTO damping at the prototype scale can be evaluated by

$$\left(C_{air,ps}^{*},{\widehat{C}}_{air,ps}^{*},{\widehat{\nu }}_{PTO,ps}^{*}\right)=\left({\displaystyle \frac{C_{air,ps}}{\rho {\omega }_{ps}\left(B_{ps}{h}_{ps}\right)V_{g,ps}}},{\displaystyle \frac{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2{C}_{air,ps}^{*}}{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2+{\left(C_{air,ps}^{*}\right)}^2}},{\displaystyle \frac{{\left(C_{air,ps}^{*}\right)}^2{\widehat{\nu }}_{PTO,\infty }^{*}}{{\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)}^2+{\left(C_{air,ps}^{*}\right)}^2}}\right).$$

(11)

Finally, with Eqs. (5), (9), and (11), the capture factor of the prototype OWC can be evaluated by

$$C_{F,ps}={\displaystyle \frac{4{\left({\left|F_E\right|}_{ms}^{*}\right)}^2\cdot \left({\widehat{\nu }}_{PTO,ps}^{*}\right)}{{\left({\widehat{C}}_{air,ps}^{*}+g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)}^2+{\left({\widehat{\nu }}_{PTO,ps}^{*}+{\nu }_{f,ms}^{*}\right)}^2}}.$$

(12)

2.2. Numerical Simulations

(a)

Computational domain and boundary conditions

In this study, three-dimensional time-domain simulations of CFD for the unsteady incompressible flows of water and air inside and outside a U-OWC were performed using the software package FLOW-3D^®, which employs the finite-volume method to solve the Reynolds-Averaged Navier–Stokes (RANS) equations. The governing equations, including the continuity equation for incompressible flow:

$$\left\{\begin{array}{l}\nabla \cdot \overrightarrow{u}=0\\ {\displaystyle \frac{\partial \overrightarrow{u}}{\partial t}}+\overrightarrow{u}\cdot \nabla \overrightarrow{u}=-{\displaystyle \frac{1}{\rho }}\nabla p+\nu {\nabla }^2\overrightarrow{u}+\overrightarrow{g}\end{array}\right.,$$

(13)

where $\overrightarrow{u}=\left(u_x,u_y,u_z\right)$ is the fluid velocity, $\nu$ the viscosity, p the pressure, and $\overrightarrow{g}$ the gravitational-acceleration, were discretized using structured computational meshes. The complex OWC geometry was precisely defined through the Fractional Area-Volume Obstacle Representation (FAVOR) technique, facilitating accurate modeling of solid boundaries within structured grids. Pressure–velocity coupling was ensured by employing the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE algorithm). The Volume-of-Fluid (VOF) method was utilized to accurately track the free surface interface between air and water, while turbulence was modeled using the k-ω turbulence closure model.

All simulations used a two-fluid incompressible approach, treating water at 20°C as Fluid 1 and air at 25°C as Fluid 2. Following the recommendation by Elhanafi et al. [31], air compressibility was neglected given its minimal impact at the selected model-scale conditions.

Figure 1 depicts the computational domain configured as a three-dimensional numerical wave tank at a 1/20th model scale. The numerical domain dimensions were three wavelengths in length ($-11.75\mathrm{m}\le x\le 0.117\mathrm{m}$), 2.4m in width ($0\mathrm{m}\le y\le 2.4\mathrm{m}$, equal to six times the OWC half-width), and 1.15m in height ($-0.5\mathrm{m}\le z\le 0.65\mathrm{m}$), with a constant water depth of 0.5m. The computational domain length was set to three wavelengths to ensure that the output data from the first three fully developed waves remained free of contamination from reflections of the wavemaker. To simplify the simulation and reduce computational requirements, symmetry conditions were applied along the x-z plane, modeling only half of the turbine geometry; thus, the measured turbine flow rates were doubled during analysis to obtain accurate results. Boundary conditions included incident wave generation (Stokes and Cnoidal waves) at the inflow boundary, symmetry at the lateral sides, solid-wall conditions at the bottom and longitudinal end, and atmospheric pressure conditions at the upper boundary (summarized in Table 1). Furthermore, a solid wall was placed immediately in front of the OWC structure to realistically simulate wave breaking.

As shown in Figure 1(a), a two-block grid arrangement was implemented to enhance computational accuracy and efficiency: Block 1 covered the whole computational domain, whereas Block 2 specifically encompassed the internal region of the U-OWC chamber. The simulation employs an adaptive time-stepping scheme to ensure stability.

A grid convergence test was conducted for Block 1 using three distinct grid sizes: 1/140, 1/150, and 1/160 of the incident wavelength (λ). The evaluation of convergence outcomes used relative differences ${\left(\mathsf{\Delta }x\%\right)}_n$ of two RAOs (Response Amplitude Operators) as defined in Eq. (14)

$$\left\{\begin{array}{l}{\left(\mathsf{\Delta }x\%\right)}_n={\displaystyle \frac{\left|{\left(x\right)}_n-{\left(x\right)}_{n-1}\right|}{\left[{\left(x\right)}_n+{\left(x\right)}_{n-1}\right]/2}}\\ RA{O}_c={\displaystyle \frac{H_c}{H_i}}\\ RA{O}_p={\displaystyle \frac{\mathsf{\Delta }p}{\rho g{H}_i}}\end{array}\right.,$$

(14)

where ${\left(\mathsf{\Delta }x\%\right)}_n$ is the relative difference between consecutive cases n and n-1,$H_i$the height of incident wave, $H_c$ the average oscillation of the water column, and $\mathsf{\Delta }p$ the air pressure drop across the PTO.

The relative difference indeed decreased as the grid size became finer. As a result, we chose the grid size of 1/160 of λ for Block 1, as it made both $\mathsf{\Delta }RA{O}_c\%$ and $\mathsf{\Delta }RA{O}_p\%$ below 1%. Block 2 was assigned a much finer grid size equal to 1/12 of the orifice diameter (0.110m) to capture detailed flow behavior within the chamber. Additionally, the water–air interface region ($-0.04\mathrm{m}\le z\le 0.04\mathrm{m}$) was refined further, with a grid size set to 1/6 of the simulated wave height to minimize numerical energy dissipation at this critical interface.

Model-scale simulations mentioned above were performed at a constant wave height of 0.075m across seven wave periods (T_ms = 1.68s, 1.79s, 1.90s, 2.12s, 2.35s, and 2.80s). As summarized in Table 2, these wave conditions represent 1:20-scale equivalents of realistic sea states, featuring full-scale wave heights of 1.5m and wave periods T_ps = 6.5s, 7.5s, 8.0s, 8.5s, 9.5s, 10.5s, and 12.5s. Based on field data documented by Chen et al. [40], these specific parameters were selected to represent the typical wave climate of coastal waters in Northeastern Taiwan—a region characterized by its high annual accumulated wave energy. This selection ensures that the numerical simulations are directly relevant to the practical design and optimization of OWCs operating in this area.

• (b) Baseline U-OWC model

Following the procedures described above, the baseline geometry of the present U-OWC, previously studied by Nguyen [41], was adopted, as illustrated in Figure 2. Table 3 presents the design parameters of the present U-OWC at T_ms = 1.90s, as reported by Nguyen [41]. These values form the basis for the current investigation and ensure consistency with prior research.

• (c) Impeller model

In this study, we used the impeller model in FLOW-3D^® to simulate the damping effect induced by the PTO (air turbine). The details of how the impeller model functions can be found in our previous work (Nguyen et al. [39]), which will be briefly described as follows.

When the axial velocity effect is neglected, the impeller model behaves as a linear turbine. Therefore, the pressure drop the impeller model induces is expressed as:

$$\mathsf{\Delta }P=\rho L{A}_d\left({\displaystyle \frac{Q}{\pi R^2}}\right),$$

(15)

where L is the thickness of the impeller, R the outer radius of the impeller, A_d the accommodation coefficient for rotational speed, and Q the net flow rate. As shown in Eq. (15), the impeller model enables precise damping control by tuning A_d, rather than changing the PTO’s dimensions and associated grid configurations as in prior research.

Figure 3 shows the schematic of the impeller model with dimeter d_o = 0.110m. To ensure satisfactory curve-fitting outcomes for $C_{F,ms}$ and the key hydrodynamic coefficients, we systematically tested different numbers of points and finally selected 11 distinct A_d values corresponding to 11 individual ${\widehat{\nu }}_{PTO,\infty }^{*}$ settings during the fitting process (see Sec. 4.1 in [39]).

3. Results and Discussion

3.1. U-OWC’s Capture Factors at Model and Prototype Scales

For the present U-OWC, across the seven examined wave periods (T_ps) ranging from 6.5s to 12.5s, all data points of capture factor yielded by the FLOW-3D^® simulations (${\widehat{\nu }}_{PTO,\infty }^{*}$, $C_{F,ms\_F3D}$) show excellent fits to the corresponding $C_{F,ms}$ curves, as illustrated in Figure 4. This result indicates the confirming the validity of the OWC mechanism described by Eq. (8). It is evident that both $C_{F,ms}$ and $C_{F,ps}$ increase with ${\widehat{\nu }}_{PTO,\infty }^{*}$ rapidly toward their respective peak values, i.e., $\max \left(C_{F,ms}\right)$ and $\max \left(C_{F,ps}\right)$, before they gradually decrease as ${\widehat{\nu }}_{PTO,\infty }^{*}$ further increases. For instance, at T_ps = 7.5s (Figure 4(b)), $C_{F,ms}$ and $C_{F,ps}$ rise sharply from zero to their respective maxima, where $\max \left(C_{F,ms}\right)=0.695$ and $\max \left(C_{F,ps}\right)=0.869$, then taper off toward zero as ${\widehat{\nu }}_{PTO,\infty }^{*}\to \infty$. Figure 4 also shows, at a given T_ps, that (1) $C_{F,ms}$ and $C_{F,ps}$ never cross each other within $0<{\widehat{\nu }}_{PTO,\infty }^{*}<\infty$, i.e., either $C_{F,ms}<C_{F,ps}$ or $C_{F,ms}>C_{F,ps}$ for the full range of ${\widehat{\nu }}_{PTO,\infty }^{*}$, and (2) $\max \left(C_{F,ms}\right)$ and $\max \left(C_{F,ps}\right)$ occur at different $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)$, i.e., the optimal value of ${\widehat{\nu }}_{PTO,\infty }^{*}$ at which $\max \left(C_{F,ms}\right)$ or $\max \left(C_{F,ps}\right)$ occurs. This variance highlights the influence of air compressibility, which not only affects the magnitude of $C_F$ but also shifts $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty }^{*}\right)$ between the model and prototype scales (details follow).

3.2. C+ and C- Intervals

Figure 5 and Table 4 present the trends and values of $\max \left(C_{F,ms}\right)$, $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)$, $\max \left(C_{F,ps}\right)$, and $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$ of the present U-OWC. It can be observed that (1) when T_ps $\le$ 7.5s, $\max \left(C_{F,ms}\right)<\max \left(C_{F,ps}\right)$ (also $C_{F,ms}<C_{F,ps}$) and $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)<\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$, and (2) when T_ps $\ge$ 8.0s, $\max \left(C_{F,ms}\right)>\max \left(C_{F,ps}\right)$ (also $C_{F,ms}>C_{F,ps}$) and $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)>\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$. For example, at T_ps = 6.5s (i.e., T_ps $\le$ 7.5s), $\max \left(C_{F,ms}\right)=0.549$ occurs at $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)=0.526$, whereas $\max \left(C_{F,ps}\right)=0.899$ occurs at $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)=0.958$; at T_ps = 9.5s (i.e., T_ps $\ge$ 8.0s), $\max \left(C_{F,ms}\right)=0.466$ occurs at $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)=0.369$, whereas $\max \left(C_{F,ps}\right)=0.390$ occurs at $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)=0.290$. These results indicate that (1) T_ps = 6.5s and 7.5s are within the C+ interval (designated by Nguyen et al. [39]) where air compressibility promotes $C_F$ and increases ${\widehat{\nu }}_{PTO,\infty }^{*}$ of the air turbine required to maximize $C_F$, and (2) T_ps = 8.0s, 8.5s, 9.5s, 10.5s, and 12.5s are within the C- interval (also designated by Nguyen et al. [39]) where air compressibility impedes $C_F$ and decreases ${\widehat{\nu }}_{PTO,\infty }^{*}$ of the air turbine required to maximize $C_F$.

At T_ps = 6.5s, the elevation of $\max \left(C_{F,ps}\right)$ (0.899) above $\max \left(C_{F,ms}\right)$ (0.549) remarkably reaches 0.35 and hence results in the highest $\max \left(C_{F,ps}\right)$ among that of other T_ps cases examined, surpassing the value (0.869) at T_ps = 7.5s with a peaking $\max \left(C_{F,ms}\right)$ (0.695). Entering the C- interval of the present U-OWC, $\max \left(C_{F,ps}\right)$ drops significantly to 0.551 at T_ps = 8.0s, and thereafter the decline of $\max \left(C_{F,ps}\right)$ below $\max \left(C_{F,ms}\right)$ mildly varies from about 0.11 to 0.08. At T_ps $\ge$ 9.5s, $\max \left(C_{F,ps}\right)$ and $\max \left(C_{F,ms}\right)$ remain almost constant around respective values of 0.38 and 0.46, i.e., low capture performance for long waves at especially the prototype scale.

, $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}\right)$, $\max \left(C_{F,ps}\right)$, and $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$, of the present U-OWC against T_ps = 6.5s, 7.5s, 8.0s, 8.5s, 9.5s, 10.5s, and 12.5s (refer to Table 2 for corresponding T_ms).

3.3. Hydrodynamic and Gravitational Coefficients

Nguyen et al. [39] demonstrated that the scaling-rematched approach developed therein enables one to obtain the OWC’s hydrodynamic and gravitational coefficients, which influence its performance as shown in Eqs. (8) and (12) for the model and prototype scales, respectively. In the present paper, we therefore analyze the performance of the present U-OWC with its dimensionless amplitude of the wave exciting force (${\left|F_E\right|}^{*}$), fluid damping coefficient (${\nu }_f^{*}$), gravitational acceleration ($g^{*}$), and, most importantly, added mass (${\mu }_A^{*}$). Figure 6 illustrates how these coefficients vary with T_ps ranging from 6.5s to 12.5s, and Table 5 provides their corresponding numerical values for detailed comparisons.

Using the linear wave theory, the amplitude of the wave exciting force $\left|F_E\right|$may be estimated as

$$\left\{\begin{array}{l}\left|F_E\right|\approx {\displaystyle {\int }_{z_{ib}}^{z_{it}}\left|p_d\right|\cdot Bdz}\\ \left|p_d\right|=\rho g\left|A_i\right|{\displaystyle \frac{\cosh k\left(z+h\right)}{\cosh kh}}\end{array}\right.,$$

(16)

where $\left|p_d\right|$ denotes the amplitude of the dynamic pressure exerted on the inlet of the water duct bounded between z_ib and z_it, and k the wavenumber. Combining Eqs. (6) and (16), ${\left|F_E\right|}^{*}$ can be shown to be approximately in inverse proportion to T_ps (in the present T_ps range) as follows

$${\left|F_E\right|}^{*}={\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{g}{h}}}\left[{\displaystyle \frac{\sinh k\left(z_{it}+h\right)-\sinh k\left(z_{ib}+h\right)}{k{V}_g\cosh kh}}\right],$$

(17)

where

$$\left\{\begin{array}{l}\sinh k\left(z_{it}+h\right)-\sinh k\left(z_{ib}+h\right)\approx k\left(z_{it}-z_{ib}\right)\sim k\\ V_g\cosh kh\approx {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{g}{\omega }}\right)\left(1+{\displaystyle \frac{2kh}{\sinh 2kh}}\right)\cosh kh={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{g}{\omega }}\right)\left({\displaystyle \frac{\sinh 2kh+2kh}{2\sinh kh}}\right)\sim {\displaystyle \frac{g}{\omega }}\sim T_{ps}\end{array}\right..$$

(18)

Finally,

$${\left|F_E\right|}^{*}\sim {\displaystyle \frac{\cancel{k}}{\cancel{k}{T}_{ps}}}\sim T_{ps}^{-1}.$$

(19)

Indeed, the approximately inverse proportionality between ${\left|F_E\right|}^{*}$and $T_{ps}$​ is found in Figure 6 for the present U-OWC. This finding provides further evidence for the reliability and robustness of the methodology proposed in this study. Furthermore, according to Eq. (16), one can notice that two factors contribute to the value of $\left|F_E\right|$ with fixed B: (1) the vertical dimension of the inlet (i.e., ${\displaystyle {\int }_{z_{ib}}^{z_{it}}dz}$), and (2) the dynamic pressure (i.e., $\left|p_d\right|$) distributed on the inlet area. Since the U-OWC usually has a shallower inlet as compared to that of the L-OWC (e.g., the L-OWC studied by Nguyen et al. [39]), the U-OWC should experience higher $\left|p_d\right|$ due to larger $\cosh k\left(z+h\right)$ and hence higher $\left|F_E\right|$ if these two types are of the same inlet’s vertical dimension.

According to Eq. (6), $g^{*}$ increases as $\omega V_g$ decreases with $Bh/S$ being constant. As shown in Figure 6 and Table 5, $g^{*}$ monotonically increases with T_ps from 0.567 at T_ps ​= 6.5s to 0.764 at T_ps ​= 12.5s, since $\omega V_g$ monotonically decreases with increasing T_ps. Again in Figure 6 and Table 5, the added-mass values ${\mu }_A^{*}$ of the present U-OWC manifest themselves to be the most influential in the performance of OWCs. Within the range of T_ps ​= 6.5s to 12.5s, ${\mu }_A^{*}$ starts with a significantly large value of 1.086 at T_ps ​= 6.5s and drops to 0.910 at T_ps ​= 7.5s. Then at T_ps = 8.0s, ${\mu }_A^{*}$ exhibits a remarkably pronounced decrease of 0.613 to 0.297, bringing the value of ${\mu }_A^{*}$ from above $g^{*}$ to below. After a little drop to 0.288 at T_ps = 8.5s, ${\mu }_A^{*}$ mildly rises to 0.387 at T_ps = 12.5s and keeps all the way below $g^{*}$. Based on Eqs. (8) and (12), the reactance associated with the hydrodynamic and gravitational coefficients is expressed in the form of $\left|g^{*}-{\mu }_A^{*}\right|$. It is obvious that (1) T_ps = 6.5s and 7.5s are within an interval coinciding with the C+ interval where $g^{*}-{\mu }_A^{*}<0$, and (2) T_ps = 8.0s, 8.5s, 9.5s, 10.5s, and 12.5s are within an interval coinciding with the C- interval where $g^{*}-{\mu }_A^{*}>0$. It is because that, at T_ps = 6.5s and 7.5s, $\left|{\widehat{C}}_{air,ps}^{*}+g_{ms}^{*}-{\mu }_{A,ms}^{*}\right|$ which represents the total reactance of the OWC system at the prototype scale (Eq. (12)) becomes less than $\left|g_{ms}^{*}-{\mu }_{A,ms}^{*}\right|$ which represents the total reactance at the model scale (Eq. (8)) since $\left(g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)$ (negative) cancels with ${\widehat{C}}_{air,ps}^{*}$ (positive); at T_ps = 8.0s, 8.5s, 9.5s, 10.5s, and 12.5s, on the other hand, $\left|{\widehat{C}}_{air,ps}^{*}+g_{ms}^{*}-{\mu }_{A,ms}^{*}\right|$ at the prototype scale becomes greater than $\left|g_{ms}^{*}-{\mu }_{A,ms}^{*}\right|$ at the model scale since $\left(g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)$ (positive) adds up with ${\widehat{C}}_{air,ps}^{*}$ (positive).

This pronounced decrease of ${\mu }_A^{*}$ for the present U-OWC may be interpreted as a phenomenon of near-resonance (when damping effects are present) with a natural period T₀, which can be roughly estimated in analogy to heave buoys using a simple formula ([42])

$$T_0\approx 2\pi \sqrt{{\displaystyle \frac{l_c}{g}}},$$

(20)

where $l_c$ denotes the length measured along the centerline of the water column from the inlet of the water duct to the free surface in the plenum chamber. Using Table 3 and Eq. (20), $l_c$ and $T_0$ of the present U-OWC can be calculated as

$$\left\{\begin{array}{l}{l}_c=\left(d+{\displaystyle \frac{e}{2}}+{\displaystyle \frac{b}{2}}+{\displaystyle \frac{a}{2}}+{\displaystyle \frac{e}{2}}+d+c\right)\times \mathrm{scale}\mathrm{ratio}\\ =\left(0.155+{\displaystyle \frac{0.14}{2}}+{\displaystyle \frac{0.155}{2}}+{\displaystyle \frac{0.17}{2}}+{\displaystyle \frac{0.14}{2}}+0.155+0.155\right)\times 20=15.35\left(\mathrm{m}\right)\\ T_0\approx 2\pi \sqrt{{\displaystyle \frac{l_c}{g}}}=2\pi \sqrt{{\displaystyle \frac{15.35}{9.81}}}=7.9\left(\mathrm{s}\right)\end{array}\right.,$$

(21)

This $T_0$ is very close to the wave period T_ps = 8.0s where ${\mu }_A^{*}$ just experiences a deep fall of 0.613 from 0.910 at T_ps = 7.5s to 0.297, indicating a strong near-resonant behavior occurring around T_ps = 8.0s.

The fluid damping coefficient ${\nu }_f^{*}$ of the present U-OWC fluctuates within a narrow range between 0.082 (T_ps = 6.5s) and 0.141 (T_ps = 9.5s), which is significantly lower than the other coefficients. Nevertheless, ${\nu }_f^{*}$ represents two of the three damping effects, in which the incoming wave energy is converted to and transferred in a specific form, i.e., the radiation and viscous damping effects stemming from ${\nu }_f^{*}$ respectively make wave to radiate away and generate heat to dissipate, and the PTO’s damping effect caused by ${\widehat{\nu }}_{PTO,\infty }^{*}$ of the air turbine takes air-flow energy off to the next-stage conversion (e.g., electricity). In other words, adding up these three forms of energy should account for the total wave energy absorbed by the OWC in operation. In order to study this aspect of hydrodynamics associated with the OWC, we define the “absorption factor” (denoted as $A_F$) following the manner of defining $C_F$ as

$$A_F={\displaystyle \frac{\mathrm{total}\mathrm{wave}\mathrm{power}\mathrm{absorbed}\mathrm{by}\mathrm{OWC}}{\mathrm{afar}\text{-}\mathrm{incoming}\mathrm{plane}\text{-}\mathrm{wave}\mathrm{power}\mathrm{within}\mathrm{OWC}\text{'}\mathrm{s}\mathrm{frontal}\text{-}\mathrm{projection}\mathrm{area}}}.$$

(22)

The value of $A_F$ characterizes the hydrodynamic energy flow entering the OWC into three modes:

(1) “attraction” mode, if $A_F>1$: the OWC attracts extra incoming wave energy from outside of its frontal-projection area possibly through certain mechanism of wave diffraction similar to the operations of near-shore devices such as BBDB (Backward Bent Duct Buoy) and OWSC (Oscillating Wave Surge Converter);

(2) “repel” mode, if $A_F<1$: the OWC repels part of the incoming wave energy out of its frontal-projection area;

(3) “harmony” mode, if $A_F=1$: the OWC is in an ideally harmonic operation with the afar-incoming (2D) plane wave.

In light of the symmetric functional structure on two damping terms in the denominator of Eq. (8) for the model scale and Eq. (12) for the prototype scale, it can be readily deduced that

$$\left\{\begin{array}{l}{A}_{F,ms}={\displaystyle \frac{4{\left({\left|F_E\right|}_{ms}^{*}\right)}^2\cdot \left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}+{\nu }_{f,ms}^{*}\right)}{{\left(g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)}^2+{\left({\widehat{\nu }}_{PTO,\infty ,ms}^{*}+{\nu }_{f,ms}^{*}\right)}^2}}=C_{F,ms}\cdot \left(1+{\displaystyle \frac{{\nu }_{f,ms}^{*}}{{\widehat{\nu }}_{PTO,\infty ,ms}^{*}}}\right)\\ A_{F,ps}={\displaystyle \frac{4{\left({\left|F_E\right|}_{ms}^{*}\right)}^2\cdot \left({\widehat{\nu }}_{PTO,ps}^{*}+{\nu }_{f,ms}^{*}\right)}{{\left({\widehat{C}}_{air,ps}^{*}+g_{ms}^{*}-{\mu }_{A,ms}^{*}\right)}^2+{\left({\widehat{\nu }}_{PTO,ps}^{*}+{\nu }_{f,ms}^{*}\right)}^2}}=C_{F,ps}\cdot \left(1+{\displaystyle \frac{{\nu }_{f,ms}^{*}}{{\widehat{\nu }}_{PTO,ps}^{*}}}\right)\end{array}\right..$$

(23)

We investigate and show in Figure 7 and Table 6 the most characteristic $A_{F,ms}$ and $A_{F,ps}$, which occur simultaneously with $\max \left(C_{F,ms}\right)$ and $\max \left(C_{F,ps}\right)$, respectively, for the seven examined T_ps ranging from 6.5s to 12.5s. It can be observed that only $A_{F,ps}\text{@}\max \left(C_{F,ps}\right)$ at T_ps = 6.5s and 7.5s are just above 1, i.e., the energy flow into the OWC is in a mild “attraction” mode, while all other values including $A_{F,ms}\text{@}\max \left(C_{F,ms}\right)$ are significantly less than 1, i.e., the energy flow into the OWC is in a strong “repel” mode. This phenomenon is due to the fact that, unlike near-shore devices as mentioned above with its wave-scattering characteristics (e.g., BBDB) or body oscillation (e.g., OWSC) to alter the incoming wave field, on-shore OWCs rarely generate wave diffraction effects because their structures are fixed to existing coastal infrastructure. Furthermore, $A_F\text{@}\max \left(C_F\right)$ appears to share the same C+ and C- intervals with $\max \left(C_F\right)$ where air compressibility respectively increases and decreases the values.

3.4. Effective Air Compressibility and PTO Damping Coefficients

Figure 8 presents the variations of ${\widehat{C}}_{air,ps}^{*}$ and ${\widehat{\nu }}_{PTO,ps}^{*}$ vs. ${\widehat{\nu }}_{PTO,\infty }^{*}$, and Table 7 shows the corresponding values of $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$, $\mathrm{opt}\left({\widehat{\nu }}_{PTO,ps}^{*}\right)$, $\mathrm{opt}\left({\widehat{C}}_{air,ps}^{*}\right)$, and $C_{air,ps}^{*}$, across wave periods T_ps​ ranging from 6.5s to 12.5s for the present U-OWC. It is important to highlight the key behaviors of these parameters, as defined in Eqs. (7) and (11), namely (1) ${\widehat{C}}_{air,ps}^{*}\to C_{air,ps}^{*}$ as ${\widehat{\nu }}_{PTO,\infty }^{*}\to \infty$ and (2) ${\widehat{C}}_{air,ps}^{*}$ and ${\widehat{\nu }}_{PTO,ps}^{*}$ intersect at $\max \left({\widehat{\nu }}_{PTO,ps}^{*}\right)=C_{air,ps}^{*}/2$ and ${\widehat{\nu }}_{PTO,\infty }^{*}=C_{air,ps}^{*}$. These characteristics are clearly illustrated in Figure 8. For instance, as shown in Figure 8(a), for T_ps = 6.5s, ${\widehat{C}}_{air,ps}^{*}\to C_{air,ps}^{*}=1.139$ as ${\widehat{\nu }}_{PTO,\infty }^{*}\to \infty$, and the curves for ${\widehat{C}}_{air,ps}^{*}$ and ${\widehat{\nu }}_{PTO,ps}^{*}$ intersect at $\max \left({\widehat{\nu }}_{PTO,ps}^{*}\right)=C_{air,ps}^{*}/2=0.570$ while ${\widehat{\nu }}_{PTO,\infty }^{*}=C_{air,ps}^{*}=1.139$.

It is evident that by varying ${\widehat{\nu }}_{PTO,\infty }^{*}$ only, the condition of complex conjugate which consists of resistance matching (i.e., ${\widehat{\nu }}_{PTO,ps}^{*}={\nu }_f^{*}$) and reactance cancelling (i.e., ${\widehat{C}}_{air,ps}^{*}+g^{*}-{\mu }_A^{*}=0$) cannot be achieved. This results in that the value of $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$ will fall between two values of ${\widehat{\nu }}_{PTO,\infty }^{*}$ which individually achieve the conditions of resistance matching or reactance cancelling. For example, at T_ps = 7.5s, the $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$ value of 0.498 indeed falls between two ${\widehat{\nu }}_{PTO,\infty }^{*}$ values of 0.134 and 0.733, which respectively satisfy the conditions of resistance matching (${\widehat{\nu }}_{PTO,ps}^{*}={\nu }_f^{*}=0.131$) and reactance cancelling (${\widehat{C}}_{air,ps}^{*}+\left(g^{*}-{\mu }_A^{*}\right)=0.330+\left(-0.330\right)=0$) as shown inTables 5 and 7.

It is also noted that there is a pair of ${\widehat{\nu }}_{PTO,\infty }^{*}$, ${\widehat{\nu }}_{PTO,\infty ,<}^{*}$ and ${\widehat{\nu }}_{PTO,\infty ,>}^{*}$, that achieve the same ${\widehat{\nu }}_{PTO,ps}^{*}$, and ${\widehat{\nu }}_{PTO,\infty ,<}^{*}<{\widehat{\nu }}_{PTO,\infty ,\mathrm{peak}}^{*}<{\widehat{\nu }}_{PTO,\infty ,>}^{*}$ where $\max \left({\widehat{\nu }}_{PTO,ps}^{*}\right)$ occurs at ${\widehat{\nu }}_{PTO,\infty }^{*}={\widehat{\nu }}_{PTO,\infty ,\mathrm{peak}}^{*}$. Across all T_ps in this study, $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$ always takes the value of ${\widehat{\nu }}_{PTO,\infty ,<}^{*}$ in the pair, which achieves $\mathrm{opt}\left({\widehat{\nu }}_{PTO,ps}^{*}\right)$. It is because that, ${\widehat{\nu }}_{PTO,\infty ,>}^{*}$ of the pair would lead to a large ${\widehat{C}}_{air,ps}^{*}$, which in turn requires a large value of $\left|g^{*}-{\mu }_A^{*}\right|$ far beyond the present data for better reactance cancelling. For example, at T_ps = 7.5s again, the pair $\left({\widehat{\nu }}_{PTO,\infty ,<}^{*},{\widehat{\nu }}_{PTO,\infty ,>}^{*}\right)$ to achieve $\mathrm{opt}\left({\widehat{\nu }}_{PTO,ps}^{*}\right)$ is (0.498 , 2.786), and ${\widehat{C}}_{air,ps}^{*}\left(2.786\right)=0.992$, which is indeed much larger than $\left|g^{*}-{\mu }_A^{*}\right|=0.330$ (see Table 5), and hence results in a worse reactance ${\widehat{C}}_{air,ps}^{*}+g^{*}-{\mu }_A^{*}=0.662>\left|g^{*}-{\mu }_A^{*}\right|$.

4. Conclusions

This study incorporates numerical simulations with the scaling-rematched approach to obtain the key hydrodynamic and air-compressibility coefficients associated with an on-shore U-OWC, namely, amplitude of the wave exciting force (${\left|F_E\right|}^{*}$), fluid damping coefficient (${\nu }_f^{*}$), gravitational acceleration ($g^{*}$), added mass (${\mu }_A^{*}$), and effective air compressibility (${\widehat{C}}_{air}^{*}$) and PTO damping coefficients (${\widehat{\nu }}_{PTO}^{*}$). The numerical simulations were conducted in the time domain using incident waves with various periods commonly observed in the Northeastern Taiwan waters. These results help elucidate the U-OWC’s hydrodynamic behavior and combined air-compressibility effects, and how they impact the capture performance under such a wave climate.

Key findings include:

• Air compressibility significantly impacts OWC performance by modifying the capture factor and optimal PTO damping from model to prototype scales. This study highlights that the U-OWC's C+ interval occurs at lower wave periods (T_ps = 6.5s and 7.5s), which is notably shorter than the C+ interval characteristic of L-OWC systems ([39]).
• The amplitude of the wave exciting force (${\left|F_E\right|}^{*}$) of the present U-OWC is approximately in inverse proportion to T_ps, consistent with a simple theory proposed in Section 3.3. This finding provides further evidence for the reliability and robustness of the methodology proposed in this study.
• The added mass values ${\mu }_A^{*}$ of the present U-OWC exhibits a remarkably pronounced decrease around T_ps = 8.0s, bringing the value of ${\mu }_A^{*}$ from above $g^{*}$ to below. Consequently, the range of negative values of $g^{*}-{\mu }_A^{*}$ at T_ps < 8.0s coincides with the C+ interval because air compressibility reduces the reactance as $\left|{\widehat{C}}_{air}^{*}+g^{*}-{\mu }_A^{*}\right|<\left|g^{*}-{\mu }_A^{*}\right|$. On the contrary, the range of positive values of $g^{*}-{\mu }_A^{*}$ at T_ps > 8.0s coincides with the C- interval because air compressibility increases the reactance as $\left|{\widehat{C}}_{air}^{*}+g^{*}-{\mu }_A^{*}\right|>\left|g^{*}-{\mu }_A^{*}\right|$.
• Following 3, the period T_ps (8.0s) around which the pronounced decrease of ${\mu }_A^{*}$ occurs is shown to be very close to the natural period (7.9s) predicted by a simple resonance formula of heave buoys, indicating that a strong near-resonance behavior occurs around this T_ps.
• The absorption factor $A_F$ is defined to be the ratio of the total wave power absorbed by the OWC via damping effects of ${\nu }_f^{*}$ and ${\widehat{\nu }}_{PTO,\infty }^{*}$ to afar-incoming plane-wave power within the OWC’s frontal-projection area. For the present U-OWC, only $A_{F,ps}\text{@}\max \left(C_{F,ps}\right)$ at T_ps = 6.5s and 7.5s are in a mild “attraction” mode (i.e., the OWC attracts a little extra incoming wave energy from outside of its frontal-projection area), while all other $A_F\text{@}\max \left(C_F\right)$ are in a strong “repel” mode (i.e., the OWC repels significant part of the incoming wave energy out of its frontal-projection area). It is because that on-shore OWCs are difficult in generating effects of wave diffraction since their bodies are fixed to existing coastal infrastructures.
• During a damping-control process, the optimal value of the PTO’s damping coefficient, $\mathrm{opt}\left({\widehat{\nu }}_{PTO,\infty ,ps}^{*}\right)$, always takes a value smaller than that which achieves $\max \left({\widehat{\nu }}_{PTO,ps}^{*}\right)$. This result highlights the critical role of air compressibility in mitigating $\left|g^{*}-{\mu }_A^{*}\right|$-induced reactance.

Future research aimed at enhancing the performance of the U-OWC will focus on increasing the front-channel depth c with two foreseeable advantages: (1) amplifying the wave-exciting force, and (2) shifting the natural period toward longer, more energetic wave regimes, thereby delaying the near-resonant effects and extending the C+ interval.