Impact of Steep Seabed Terrains on Oscillating Buoy-Wave Energy-Converter Performance

Abstract

This paper employs Computational Fluid Dynamics (CFD) methods to develop a numerical model of an oscillating buoy-wave energy converter and investigates the impact of steep seabed topography near islands and reefs on its performance. The model’s accuracy is validated by comparison with experimental results from the published literature. Subsequently, the influence of deployment location, reef-front slope gradient, and reef-flat water depth on the device’s performance is analyzed. The results indicate that the strategic utilization of steep seabed topography can significantly enhance the energy capture efficiency of the device in long-wave regions. This study provides valuable references for the design and deployment of oscillating buoy-wave energy converters in near-reef areas.

1. Introduction

Rising energy demand and environmental pollution have heightened the global focus on renewable energy development [1]. Among various clean energy sources, wave energy has garnered significant attention due to its high energy density, low environmental impact, and wide distribution [2]. Wave energy converters (WECs) can be classified based on their operating principles into oscillating-water-column wave energy converters (OWCWECs), overtopping wave energy converters (OWECs), and oscillating buoy-wave energy converters (OBWECs) [3]. Currently, research on oscillating buoy-wave energy devices is extensive. In addition to physical model tank experiments, numerical simulation has become an effective means of studying wave energy devices due to its low cost, high flexibility, and ease of data acquisition [4]. Potential flow theory was initially applied in marine engineering and related research fields. Cho et al. systematically investigated the hydrodynamic performance of a two-concentric-cylindrical-body wave energy converter using a combination of analytical solutions and model tests, proposing several design strategies to improve the Power Take-Off (PTO) energy-conversion efficiency [5]. Zhang et al. established a mechanical model using Newton’s second law and Airy wave theory to systematically study the hydrodynamic behavior of multiple buoys in the Sharp Eagle wave energy converter, analyzing the coupling motion of the buoys under the influence of incident waves [6]. Ma et al. used AQWA to establish a heave motion model for a twin-hull floating-point absorber (FPA) WEC. They analyzed the device’s frequency-domain hydrodynamic performance and time-domain motion characteristics and studied the effect of periods on the conversion efficiency of the twin-hull FPA device [7].

The application of CFD methods in handling strongly nonlinear phenomena has enhanced the accuracy of wave-energy-device calculations. Zhang et al. utilized Star CCM+ to develop a two-dimensional numerical model to analyze the hydrodynamic characteristics of a combined oscillating buoy-wave energy device and floating breakwater system. By examining the vortex field around the device, they discovered that asymmetrical shapes possess higher power-conversion efficiency and better wave attenuation performance [8]. Sun et al. employed Fluent to model the interaction between waves and both flap-type and Sharp Eagle-type oscillating surge wave energy converters (OSWECs), focusing on the nonlinear phenomena of overtopping and slamming [9]. Schmitt et al. assessed the applicability of the Reynolds-averaged Navier–Stokes solver in OpenFOAM for simulating oscillating wave surge converters (OWSCs) and found good agreement with experimental results [10]. Luan et al. used CFD methods to establish a three-dimensional numerical wave tank to study the hydrodynamic performance of a single-buoy WEC under different wave conditions. The results indicated that the buoy’s motion response and wave force are related to the PTO damping constant, which does not affect the wave-number turning point of the optimal damping constant [11].

Most of the aforementioned studies focus on the interaction between waves and devices over a flat seabed, with few addressing the effects of seabed topography on wave energy devices. Rezanejad et al. developed a two-dimensional numerical model in OpenFOAM to investigate the primary capture efficiency of an OWC under stepped seabed conditions. They demonstrated that a stepped seabed can effectively enhance capture efficiency [12]. Mohapatra et al. used CFD methods to create a wave tank for studying the performance of an OWC over an inclined seabed, finding that a sloped seabed significantly increases the capture efficiency of floating OWCs, although this effect varies with the period of the incident waves [13]. Gao et al. utilized the CFD software OpenFOAM(OpenFOAM® version 3.0.1) to examine the phenomenon of free surface resonance within narrow gaps and to simulate the impact of topographical variations on this resonance. Their findings indicate that the resonant frequency decreases monotonically with increasing slope, while the amplification of resonant wave height and the reflection coefficient display a fluctuating pattern as the slope changes [14]. Belibassakis et al. established a numerical model to predict the performance of wave energy devices in areas with varying water depths, focusing on how buoy properties under non-uniform depth conditions impact device performance [15]. Mandal et al. applied the Fourier–Bessel series expansion method and the matched characteristic-function expansion method to research the effect of a stepped sea bottom on the heave response operator of a heaving dual-concentric-cylindrical wave-energy-converter buoy. Their results showed that stepped seabed topography can enhance wave energy capture efficiency to some extent [16]. Zhang et al., based on linear potential flow theory, examined the hydrodynamic behavior of a pendulum-type WEC array installed on a stepped seabed. Their findings indicated that quarter-wavelength resonance on a stepped seabed topography improved hydrodynamic efficiency, but efficiency declined beyond the critical wavenumber, accompanied by the emergence of Rayleigh–Bloch waves [17].

Some of the studies mentioned have already highlighted the impact of uneven seabed topography on wave-energy-device performance. However, there is a relative scarcity of research specifically addressing the effects of steep and variable seabed topography, characteristic of near-island and reef regions, on oscillating buoy-wave energy converters. The complex and abrupt changes in water depth and intricate seabed structures in these areas significantly influence wave propagation, reflection, refraction, and energy distribution [18], potentially exerting substantial effects on the performance of wave energy devices. Moreover, the sharply varying seabed topography near islands and reefs presents challenges for anchoring floating devices [19]. However, near-shore bottom-fixed devices offer new avenues for wave energy utilization in these regions.

This paper focuses on a bottom-fixed OBWEC designed for near-island and reef coastal areas, aiming to provide clean electricity for remote islands. Bottom-fixed devices offer better solutions to anchoring challenges but must account for the significant impact of rapidly changing seabed topography on WEC performance. To address this, a three-dimensional numerical model is developed using CFD methods. The model’s accuracy is validated by comparing it with experimental results from Mandal et al. [14]. The model incorporates a linear spring-damping system articulated separately with the buoy and underwater attachment to simulate practical hydraulic PTO systems. Based on this, this study investigates the effects of different deployment locations, reef-front slope gradients, and reef-flat water depths on device performance.

2. Numerical Model

This study establishes a numerical wave tank based on the incompressible Navier–Stokes equations for a Newtonian fluid, employing the volume of fluid (VOF) method to capture the free surface motion. The viscous terms are modeled as described in [20], utilizing the k-ε turbulence model. The computational domain is depicted in Figure 1, simulating island reef topography with a simplified slope and a platform behind it. No-slip boundary conditions are applied to all surfaces of the structures and the seabed. The Cartesian coordinate system (o, x, y, z) is defined with the origin o, positioned at the center of the water plane of the submerged body. In this system, the x-axis aligns with the direction of wave propagation, the y-axis represents the width of the numerical wave tank, and the z-axis extends vertically upward. SWL denotes the still water level. The total length of the computational domain in the x-direction is 6λ (where λ denotes the wavelength). The water depth in the deep-sea region is fixed at H = 50 m, and the water depth over the reef flat is h. The horizontal distance between the buoy’s centroid and the reef edge is X, the buoy’s length is L, and the slope gradient in front of the reef is $m=k:l$, where k represents the height of the slope and l denotes the horizontal distance along the slope. An air region 20 m high is situated above the water surface, and the total width of the computational domain is four times the buoy’s geometric width. The computational domain is divided into three regions in the x-direction: the middle section serves as the working zone, while the front and rear ends are wave-generation and wave-absorption zones, respectively, each 1.5λ in length, to ensure the stability of the incident waves.

2.1. Fundamental Governing Equations

In the model, the equations describing the motion of Newtonian incompressible viscous fluids are the continuity equation and the Reynolds-averaged Navier–Stokes equations [21], presented as follows:

(1)$\nabla \cdot u=0$

(2)${\displaystyle \frac{\partial u}{\partial t}}+\left(u\cdot \nabla \right)u=\mu {\nabla }^{2}u-{\displaystyle \frac{1}{\rho }}\nabla p+g$

In the equation, $u=\left(u,v,w\right)$ represents the velocity vector of the fluid; $t$ represents time; $\rho$ represents the density of the fluid; $\mu$ represents the dynamic viscosity of the fluid; $p$ represents the pressure of the fluid; and $g$ represents the acceleration due to gravity.

2.2. Volume of Fluid (VOF) Method

Considering the gas–liquid two-phase flow, the model employs the VOF method to capture the movement of the free liquid surface [22]. The specific governing equations are as follows:

(3)${\displaystyle \frac{\partial \gamma }{\partial t}}+\nabla \cdot \left(u\gamma \right)+\nabla \cdot \left[u_r\gamma \left(1-\gamma \right)\right]=0$

In the equation, $\gamma$ denotes the volume fraction of water; if $\gamma =1$, it signifies that the fluid within the grid is entirely water, if $0<\gamma <1$, it indicates a mixture of water and gas within the grid, and $\gamma =0$ denotes that the grid contains only gas. ${\mathbf{u}}_{\mathbf{r}}$ represents the relative velocity between the liquid phase and the gas phase. The physical properties of the fluid within the grid can be expressed by the following equation:

(4)$\rho =\gamma {\rho }_{water}+\left(1-\gamma \right){\rho }_{air}$

(5)$\mu =\gamma {\mu }_{water}+\left(1-\gamma \right){\mu }_{air}$

In the equation, $\rho$ denotes the density of the fluid, and $\mu$ represents the dynamic viscosity. The subscripts “water” and “air”, respectively, indicate the properties related to water and air.

2.3. Motion and Energy Conversion of Buoys

Figure 2 illustrates the motion model of the device. Prior research on the Sharp Eagle wave energy converter has been extensive [6]. The device primarily consists of three components: the Eagle-head absorber float, the underwater attachment, and the hydraulic PTO system. The Eagle-head absorber float and the underwater attachment experience relative motion under the influence of waves, driving the hydraulic cylinder to convert captured wave energy into hydraulic energy stored in the energy storage system. This hydraulic energy is subsequently converted into electrical energy by driving an electric generator with a hydraulic motor. The model is based on Newton’s second law, solving the object’s trajectory based on the forces acting on the float. The underwater attachment is fixed to the reef flat, and the Eagle-head absorber float is connected to the underwater attachment in a hinged manner, allowing for single-degree-of-freedom pitch motion around the hinge point O. A linear spring-damping system, hinged at points B and A to the underwater attachment and the absorber float, respectively, is included in the model to simulate the hydraulic PTO system.

The equations of motion are as follows:

(6)$M_o=M_g+M_f+M_{pto}$

(7)$M_o=I\dot{\omega }$

(8)$M_g+M_f=I\ddot{\beta }$

(9)$M_{pto}=-F_{pto}{L}_{AO}\cos \alpha$

(10)$I\dot{\omega }=I\ddot{\beta }-F_{pto}{L}_{AO}\cos \alpha$

(11)$F_{pto}=C_{pto}{\dot{L}}_{AB}$

where $M_o$ donates the total moment exerted on the buoy; $M_g$ is the moment generated by gravity; $M_f$ is the moment generated by the fluid forces; $M_{pto}$ is the moment generated by the PTO; $I$ is the rotational inertia of the buoy; $\omega$ is the angular velocity of the buoy under load; $\beta$ is the angular displacement of the buoy when unloaded; $F_{pto}$ is the damping force of the hydraulic cylinder; $C_{pto}$ is the damping coefficient of the PTO; and $L_{AO}$ and $L_{AB}$ represent the distances from point A to point O and from point A to point B, respectively.

The expressions for the average output of the PTO device and the capture width ratio (CWR) [23] are as follows:

(12)$P_{pto}=C_{pto}{\displaystyle {\int }_0^{t_0}{{\dot{L}}_{AB}}^2}\mathrm{dt}$

(13)$P_{wave}={\displaystyle \frac{\rho g{H_i}^2\omega }{16k}}\left(1+{\displaystyle \frac{2kh}{\sinh 2kh}}\right)b{t}_0$

(14)$CWR={\displaystyle \frac{P_{pto}}{P_{wave}}}$

where $P_{pto}$ is the average output power of the PTO; $P_{wave}$ is the incident wave power; $\rho$ is the density of the fluid; $H_i$ is the wave height of the incident wave; $\omega$ is the angular frequency of the incident wave; $k$ is the wave number; and $b$ represents the width of the wave-absorbing float facing the wave.

3. Study of Grid Independence and Validation of Model Accuracy

3.1. Study of Grid Independence

When using a CFD numerical wave tank for simulation, it is crucial to perform grid and time-step independence verification to balance computational cost and result accuracy [24]. For this study, the systematic verification of the grid spacing in the z-direction, $\Delta z$, and in the x-direction, $\Delta x$, at the water free surface was carried out. A typical sea condition in the South China Sea was selected, characterized by a period T of 5.5 s, a wave height $H_i$ of 1.0 m, and a water depth of 50 m [25]. Three different time steps and three grid sizes were chosen, with specific data shown in Table 1. Figure 3a presents the temporal evolution curves at the center of the tank for the three different time steps, showing significant differences between Case 1, Case 2, and Case 3. Table 2 compares the differences in wave height from the theoretical values for these three different time steps, indicating that the error in Case 1 reaches 7%, while Case 2 and Case 3 have errors of only 3.7% and 2.7%, respectively. Considering computational cost, the time step set for Case 2’s model meets the accuracy requirements.

A grid convergence study was conducted based on the aforementioned time steps. Figure 3b shows the temporal evolution curves of the wave surface at the center of the tank for Case 2, Case 4, and Case 5. It can be observed that Case 2 and Case 5 are closely aligned, while Case 4 shows a significant difference from the other two. Table 3 provides the differences in wave height from the theoretical values for the three different grids. Therefore, for the subsequent studies in this paper, the grid and time-step settings of the Case 2 model were chosen [26].

3.2. Model Accuracy Verification

Mandal et al. studied the performance of a dual-cylinder floating wave energy device on stepped seabed terrain [16]. To validate the accuracy of the numerical model, this paper simulated the model from Mandal et al. and compared the results with their experimental findings. Figure 4 shows the variation of the relative heave RAO of the dual floats with the incident wave period, comparing the simulation results from this paper to the experimental results. “NS” denotes no seabed step terrain, while “WS” denotes the presence of seabed step terrain. It can be seen that the results from the numerical model used in this paper are in good agreement with the experimental results.

4. Results Evaluation and Discussion

The island reef topography is depicted in Figure 1. According to Meng et al.’s research [27], the Nansha Islands are predominantly characterized by large atolls, with the water depth of their reef flats mostly ranging from 3 to 10 m. Beyond the reef edge, the water depth increases sharply, and the slope of the reef front is steep, often exceeding 60°, and is sometimes vertical. In this section, based on the characteristics of the device and the terrain, the impact of different deployment positions (X = −2L, L, 0, L, 2L), different reef-front slope angles (m = 1:1, 2:1, vertical), and different reef-flat water depths (h = 6 m, 8 m, 10 m) on the device’s performance is studied and compared with the condition without terrain. According to the data [25], the wave periods frequently encountered during operations in the South China Sea range from 3 to 8 s. Therefore, the incident wave period is set between 3 and 8 s, and the wave height is set to 2 m for the following calculations.

4.1. Impact of Linear PTO Damping Coefficient on Wave Energy Device

Figure 5 shows the variation of the average output power of the device’s PTO with the linear damping coefficient for different periods, while Figure 6 displays the variation of the heave motion amplitude of the absorber float under no load with the incident wave period. From Figure 6, it can be observed that as the incident wave period increases, the heave motion amplitude of the absorber float first increases and then decreases, with the peak occurring at approximately 4.7 s, which is the resonant period of the float. On the other hand, from Figure 5, it can be seen that the average output power of the PTO under different periods first increases and then decreases with an increase in the damping coefficient, with the peak corresponding to the optimal PTO damping coefficient for that wave condition. Specifically, as shown in Figure 5a, when the wave period is less than the resonant period (T = 4.7 s), the optimal PTO damping coefficient decreases with an increase in the incident wave period. Conversely, as shown in Figure 5b, when the incident wave period is greater than the resonant period, the optimal PTO damping coefficient increases with an increase in the incident wave period.

In conclusion, wave energy devices require different optimal PTO damping coefficients for different sea conditions to maximize capture efficiency. However, continuously adjusting the PTO damping coefficient to adapt to varying sea conditions is challenging in practical engineering. Therefore, it is necessary to select a fixed PTO damping coefficient based on actual engineering requirements to achieve relatively optimal performance. According to data from Wan et al., the actual working sea conditions have wave periods concentrated between 4 and 6 s. Based on Figure 5, $C_{PTO}=6.0\times {10}^5$ is selected as the fixed damping coefficient for the device. This fixed damping coefficient will be used as the basis for subsequent calculations.

4.2. Influence of Device Deployment Positions

Figure 7 illustrates the variation of the device’s PTO average output power and the pitch amplitude of the absorber float with the incident wave period; the device is placed at positions X = −2L, X = −L, X = 0, X = L, and X = 2L relative to the reef edge. The model parameters at this time are a reef-flat water depth of h = 8 m and a reef-front slope ratio of m = 2:1. It can be observed that as the incident wave period increases, the PTO average output power of the device at different deployment positions first increases and then decreases, with the peak power occurring at an incident wave period of approximately 5 s. Specifically, when the device is placed at X = 0, the PTO average output power reaches its highest point at 38.76 kW.

Moreover, between incident wave periods of 4 to 7 s, the device maintains higher PTO average output power when deployed at position X = 0. As the device moves backward towards the reef flat to positions X = L and X = 2L, both the PTO average output power and the pitch amplitude of the float sequentially decrease. Figure 8 illustrates the temporal evolution curves for an incident wave period of 7 s, showing a significant deviation in the damping-force curves at X = L and X = 2L. Although the peaks remain at relatively high levels, the curves narrow overall. This occurs primarily because wave energy begins to dissipate continuously as waves enter the shallow-water area and encounter bottom friction and the shallow deformation of waves [28].

We now consider the device’s performance when placed in front of the reef edge at positions X = −L and X = −2L. Firstly, when the device is deployed at X = −L, the PTO average output power is lower than at X = 0. In Figure 8, the temporal damping-force curves at these two positions largely overlap, but the peak at X = −L is lower. When the device is positioned at X = −2L, the PTO average output power is relatively high at an incident wave period of 5 s, but it rapidly decreases with the increasing incident wave period, as shown in Figure 8, where the peak at X = −2L is notably smaller. This trend may result from the convergence of wave energy in the vertical direction as waves propagate from the deep sea to the reef, where the water depth transitions from deep to shallow. Consequently, wave energy gradually increases as it propagates up the reef-front slope, reaching its maximum at the reef edge within a certain period range.

4.3. Influence of Different Terrain Slopes on Device Performance

In Section 4.2, we investigate the impact of device placement on performance and determine that optimal performance occurs when the float center aligns with the vertical plane of the reef edge. Expanding on this, we examined the variation of PTO output power and float heave amplitude with the incident wave period under four different terrain conditions, where WS, $\gamma =1$, $\gamma =2$, and $\gamma =3$, respectively, represent no terrain, vertical step terrain, m = 2:1, and m = 1:1 The model parameters are set as X = 0 and h = 8 m. From Figure 9, it is evident that as the incident wave period increases, the performance of the device under the four terrain conditions can be roughly divided into three stages. In the first stage (incident wave period is less than 4 s), where wave lengths are shorter and terrain impact on wave propagation is minimal, PTO average output power and float pitch amplitude show consistent values across all conditions. In the second stage (incident wave period between 4–6 s), longer wave lengths and terrain characteristics begin to influence wave reflection and refraction, resulting in slightly higher PTO average output power and float heave amplitudes under the no-terrain conditions compared to terrains. Moving to the third stage (incident wave period greater than 6 s), where longer wave lengths intensify terrain effects, wave energy converges vertically, significantly boosting device output power under terrain conditions—an observation to be further validated in the subsequent section. Additionally, across varying reef-front slope ratios, the device’s PTO average output power and float pitch amplitude remain consistent, suggesting the minimal influence of the reef slope on performance. Nevertheless, for subsequent analysis, the terrain condition showing slightly superior performance will be selected as the reference.

4.4. Influence of Different Reef-Flat Depths on Device Performance

As the incident wave period increases, the performance of the device under different terrain conditions exhibits three stages of variation. This section explores the impact of reef-flat depth on device performance. Figure 10 illustrates the device’s performance under four reef-flat depth conditions, with model parameters set at X = 0 m and the slope ratio set at m = 2:1.

In the first stage (incident wave period less than 4 s), regardless of varying reef-flat depths, the device’s PTO average output power and float pitch amplitude are basically the same. Moving into the second stage (incident wave period between 4–6 s), optimal performance occurs when the reef-flat depth is 6 m, surpassing the condition without terrain (h = 50 m). This suggests that shallower depths intensify the terrain’s influence on the wave dynamics, potentially accelerating entry into the third stage. In the third stage (incident wave period greater than 6 s), performance under the h = 10 m condition exceeds that without terrain only when the incident wave period approaches 7 s. This implies that deeper reef-flat depths may delay the onset of the second stage, thereby requiring longer periods (wavelengths) for the device to surpass the performance without terrain.

As depicted in Figure 11, at an incident wave period of 7 s, the damping-force curve peaks decrease with increasing reef-flat depth. This indicates that shallower reef-flat depths may enhance the terrain’s impact on wave behavior, facilitating an earlier transition into the third stage. Conversely, deeper reef-flat depths may prolong the duration of the second stage, resulting in the comparatively subdued performance of the device under terrain conditions.

4.5. Study on Wave Energy-Conversion Efficiency

Through a comprehensive analysis of the impact of steep seabed terrain on wave-energy-device performance, we identified optimal terrain conditions (X = 0, m = 2:1, h = 6 m) and compared their capture width ratios (CWRs) with conditions lacking terrain. As depicted in Figure 12, the results indicate that in the short-wave region (T < 4 s), both conditions exhibit similar capture widths. However, in the long-wave region (T = 4–8 s), the device’s efficiency under terrain conditions notably surpasses that without terrain. For instance, at an incident wave period of 6 s, the device achieves a capture width of 35.6% under terrain conditions, marking an 18.6% increase compared to conditions lacking terrain. This underscores the effective enhancement of wave condition adaptability through the utilization of steep seabed terrain.

5. Conclusions

This study employed a CFD model to explore how a steep seabed terrain affects the performance of an oscillating buoy-wave energy converter. After analyzing the optimal PTO damping coefficient, a suitable fixed value was chosen for subsequent calculations. This study comprehensively assessed the device’s performance across various deployment positions, reef-slope gradients, and reef-flat depths. The findings highlight the following conclusions:

The device exhibits optimal performance when deployed at the reef edge (X = 0), as indicated by the analysis of its performance at four deployment positions. Moving the device backward along the reef (positions X = L and X = 2L) resulted in reduced average output power from the PTO and a decrease in the pitch amplitude of the float. Similarly, placing the device in front of the reef edge yielded inferior performance compared to positioning it directly at the reef edge.

When comparing the device’s performance with and without terrain, it was observed that as the incident wave period increases, both the PTO average output power and the pitch amplitude of the float exhibit trends that can be categorized into three stages. In the short-wave region (incident wave period less than 4 s), the terrain has minimal influence on wave propagation, resulting in comparable performance with and without terrain. In the medium-wavelength region (incident wave period of 4–6 s), the device’s performance with terrain slightly lags behind that without terrain. However, in the long-wave region (incident wave period greater than 6 s), the presence of terrain significantly enhances the device’s performance. The influence of different reef slope gradients on the device’s performance was found to be insignificant.

Further research indicates that altering the depth of the reef flat significantly impacts wave-energy-device performance. Shallower reef depths enhance the device’s ability to capture wave energy. Under optimal terrain conditions (X = 0, m = 2:1, h = 6 m), the device’s capture width remains comparable to that under no-terrain conditions in the short-wave region but is notably higher in the long-wave region. This underscores how effectively utilizing steep seabed terrain can enhance the device’s adaptability to various wave conditions.

This study examines the impact of steep seabed topography in near-island reef areas on the performance of an OBWEC. The results indicate that strategically utilizing this topography can significantly improve its performance. These findings offer essential guidance for the design and deployment of wave energy converters in shallow and nearshore environments, emphasizing the need to optimize both design and site selection based on local seabed characteristics.

Finally, we declare that these conclusions are applicable only to the range of incident wave conditions, the variations in seabed topography (including terrain slope and reef-flat depth), and the geometric configuration of the buoy described in this paper.

Author Contributions

Conceptualization, Z.W. and C.L.; methodology, Z.W.; software, C.L. and M.C.; validation, S.S. and X.Y.; formal analysis, Z.W.; investigation, C.L.; resources, S.S.; data curation, W.W.; writing—original draft preparation, C.L.; writing—review and editing, W.W.; visualization, M.C.; supervision, S.S; project administration, S.S.; funding acquisition, Z.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data presented in this study are available on request from the corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1. Schematic diagram of the computational domain of the numerical model.

Figure 2. Motion principle diagram of Sharp Eagle wave energy device.

Figure 3. Time history curve of wave height at the center of the computational domain: (a) Convergence study for different time steps; (b) Convergence study with mesh size.

Figure 4. Variation of the relative heave RAO of the dual floats with the incident wave period.

Figure 5. Average output power of PTO under different periods: (a) T = 3–4.5 s; (b) T = 5–8 s.

Figure 6. Amplitude of the absorber float’s heave motion under no load.

Figure 7. Variation of PTO average output power and float heave amplitude at different deployment positions with incident wave period: (a) PTO average output power; (b) Pitch amplitude of the floater.

Figure 8. Temporal evolution curves of PTO damping force at T = 7 s.

Figure 9. Variation of PTO average output power and float heave amplitude with incident wave period under different slope ratios and no-terrain conditions: (a) PTO Average output power; (b) Pitch amplitude of the floater.

Figure 10. Variation of PTO average output power and float heave amplitude with incident wave period under different reef-flat depth conditions: (a) PTO Average output power; (b) Pitch amplitude of the floater.

Figure 11. Temporal evolution of PTO damping force at T = 7 s.

Figure 12. Variation of capture width ratio with incident wave period.

Table 1

Grid size and time step parameters.

Case	Mesh Size	Time Steps	Total Number of Elements
1	$\Delta z=H_i/20; \Delta x=\lambda /80$	$\Delta t=T/500$	1,031,322
2	$\Delta z=H_i/20; \Delta x=\lambda /80$	$\Delta t=T/1000$	1,031,322
3	$\Delta z=H_i/20; \Delta x=\lambda /80$	$\Delta t=T/1500$	1,031,322
4	$\Delta z=H_i/10; \Delta x=\lambda /40$	$\Delta t=T/1000$	352,350
5	$\Delta z=H_i/40; \Delta x=\lambda /160$	$\Delta t=T/1000$	3,078,138

Table 2

Comparison of wave heights with different time step parameters.

Case	Wave Height/m	Relative Error/%	Wave Period/s	Relative Error/%
1	0.930	7	5.495	0.09
2	0.963	3.7	5.497	0.05
3	0.973	2.7	5.496	0.07

Table 3

Comparison of wave heights with different grid sizes.

Case	Wave Height/m	Relative Error/%	Wave Period/s	Relative Error/%
2	0.963	3.7	5.497	0.05
4	0.954	4.6	5.493	0.12
5	0.972	2.8	5.497	0.05

References

1. Lu, Y.; Khan, Z.; Alvarez-Alvarado, M.; Zhang, Y.; Huang, Z.; Imran, M. A Critical Review of Sustainable Energy Policies for the Promotion of Renewable Energy Sources. Sustainability; 2020; 12, 5078. [DOI: https://dx.doi.org/10.3390/su12125078]

2. Zhang, Y.; Zhao, Y.; Sun, W.; Li, J. Ocean Wave Energy Converters: Technical Principle, Device Realization, and Performance Evaluation. Renew. Sustain. Energy Rev.; 2021; 141, 110764. [DOI: https://dx.doi.org/10.1016/j.rser.2021.110764]

3. Falcao, A. Wave Energy Utilization: A Review of the Technologies. Renew. Sustain. Energy Rev.; 2010; 14, pp. 899-918. [DOI: https://dx.doi.org/10.1016/j.rser.2009.11.003]

4. Palm, J.; Eskilsson, C.; Moura Paredes, G.; Bergdahl, L. Coupled Mooring Analysis for Floating Wave Energy Converters Using CFD: Formulation and Validation. Int. J. Mar. Energy; 2016; 16, pp. 83-99. [DOI: https://dx.doi.org/10.1016/j.ijome.2016.05.003]

5. Cho, I.; Kim, M. Hydrodynamic Performance Evaluation of a Wave Energy Converter with Two Concentric Vertical Cylinders by Analytic Solutions and Model Tests. Ocean Eng.; 2017; 130, pp. 498-509. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2016.11.069]

6. Zhang, Y.; Sheng, S.; You, Y.; Huang, Z.; Wang, W. Study of Hydrodynamic Characteristics of a Sharp Eagle Wave Energy Converter. China Ocean Eng.; 2017; 31, pp. 364-369. [DOI: https://dx.doi.org/10.1007/s13344-017-0043-0]

7. Ma, Y.; Zhang, A.; Yang, L.; Li, H.; Zhai, Z.; Zhou, H. Motion Simulation and Performance Analysis of Two-Body Floating Point Absorber Wave Energy Converter. Renew. Energy; 2020; 157, pp. 353-367. [DOI: https://dx.doi.org/10.1016/j.renene.2020.05.026]

8. Zhang, H.; Zhou, B.; Vogel, C.; Willden, R.; Zang, J.; Zhang, L. Hydrodynamic Performance of a Floating Breakwater as an Oscillating-Buoy Type Wave Energy Converter. Appl. Energy; 2020; 257, 113996. [DOI: https://dx.doi.org/10.1016/j.apenergy.2019.113996]

9. Sun, L.; Yang, Z.; Chen, M.; Li, F. Analysis of Sharp Eagle Oscillating Surge Wave Energy Converter Based on a Two-Dimensional Numerical Wave Flume Model. J. Mar. Sci. Eng.; 2023; 11, 1607. [DOI: https://dx.doi.org/10.3390/jmse11081607]

10. Schmitt, P.; Elsaesser, B. On the Use of OpenFOAM to Model Oscillating Wave Surge Converters. Ocean Eng.; 2015; 108, pp. 98-104. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2015.07.055]

11. Luan, Z.; He, G.; Zhang, Z.; Jing, P.; Jin, R.; Geng, B.; Liu, C. Study on the Optimal Wave Energy Absorption Power of a Float in Waves. J. Mar. Sci. Eng.; 2019; 7, 269. [DOI: https://dx.doi.org/10.3390/jmse7080269]

12. Rezanejad, K.; Gadelho, J.; Soares, C. Hydrodynamic Analysis of an Oscillating Water Column Wave Energy Converter in the Stepped Bottom Condition Using CFD. Renew. Energy; 2019; 135, pp. 1241-1259. [DOI: https://dx.doi.org/10.1016/j.renene.2018.09.034]

13. Mohapatra, P.; Bhattacharyya, A.; Sahoo, T. Performance of a Floating Oscillating Water Column Wave Energy Converter over a Sloping Bed. Ships Offshore Struct.; 2021; 16, pp. 659-669. [DOI: https://dx.doi.org/10.1080/17445302.2020.1772665]

14. Gao, J.; He, Z.; Zang, J.; Chen, Q.; Ding, H.; Wang, G. Topographic Effects on Wave Resonance in the Narrow Gap between Fixed Box and Vertical Wall. Ocean Eng.; 2019; 180, pp. 97-107. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2019.03.040]

15. Belibassakis, K.; Bonovas, M.; Rusu, E. A Novel Method for Estimating Wave Energy Converter Performance in Variable Bathymetry Regions and Applications. Energies; 2018; 11, 2092. [DOI: https://dx.doi.org/10.3390/en11082092]

16. Mandal, S.; John, A.; Kumar, V. Performance of a Two-Body Heaving Cylindrical Wave Energy Converter on Stepped Sea Bottom. Ocean Eng.; 2023; 287, 115762. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2023.115762]

17. Zhang, Y.; Zhu, W.; Cheng, X. Wave Power Extraction of Flap-Type Wave Energy Converter Array Mounted at the Stepped Bottom Topography. Renew. Energy; 2023; 218, 119334. [DOI: https://dx.doi.org/10.1016/j.renene.2023.119334]

18. Sheremet, A.; Kaihatu, J.; Su, S.; Smith, E.; Smith, J. Modeling of Nonlinear Wave Propagation over Fringing Reefs. Coast. Eng.; 2011; 58, pp. 1125-1137. [DOI: https://dx.doi.org/10.1016/j.coastaleng.2011.06.007]

19. Huang, S.; Sheng, S.; Gerthoffert, A.; Cong, Y.; Zhang, T.; Wang, Z. Numerical Design Study of Multipoint Mooring Systems for the Floating Wave Energy Converter in Deep Water with a Sloping Bottom. Renew. Energy; 2019; 136, pp. 558-571. [DOI: https://dx.doi.org/10.1016/j.renene.2019.01.027]

20. Li, Y.; Lin, M. Regular and Irregular Wave Impacts on Floating Body. Ocean Eng.; 2012; 42, pp. 93-101. [DOI: https://dx.doi.org/10.1016/j.oceaneng.2012.01.019]

21. Windt, C.; Davidson, J.; Ringwood, J. High-Fidelity Numerical Modelling of Ocean Wave Energy Systems: A Review of Computational Fluid Dynamics-Based Numerical Wave Tanks. Renew. Sustain. Energy Rev.; 2018; 93, pp. 610-630. [DOI: https://dx.doi.org/10.1016/j.rser.2018.05.020]

22. Hirt, C.; Nichols, B. Volume Of Fluid (Vof) Method For The Dynamics Of Free Boundaries. J. Comput. Phys.; 1981; 39, pp. 201-225. [DOI: https://dx.doi.org/10.1016/0021-9991(81)90145-5]

23. Price, A.A.E.; Dent, C.J.; Wallace, A.R. On the Capture Width of Wave Energy Converters. Appl. Ocean Res.; 2009; 31, pp. 251-259. [DOI: https://dx.doi.org/10.1016/j.apor.2010.04.001]

24. Windt, C.; Davidson, J.; Ransley, E.; Greaves, D.; Jakobsen, M.; Kramer, M.; Ringwood, J. Validation of a CFD-Based Numerical Wave Tank Model for the Power Production Assessment of the Wavestar Ocean Wave Energy Converter. Renew. Energy; 2020; 146, pp. 2499-2516. [DOI: https://dx.doi.org/10.1016/j.renene.2019.08.059]

25. Yaakob, O.; Hashim, F.E.; Omar, K.M.; Din, A.H.M.; Koh, K.K. Satellite-Based Wave Data and Wave Energy Resource Assessment for South China Sea. Renew. Energy; 2016; 88, pp. 359-371. [DOI: https://dx.doi.org/10.1016/j.renene.2015.11.039]

26. Khan, A.; Irfan, M.; Niazi, U.; Shah, I.; Legutko, S.; Rahman, S.; Alwadie, A.; Jalalah, M.; Glowacz, A.; Khan, M. Centrifugal Compressor Stall Control by the Application of Engineered Surface Roughness on Diffuser Shroud Using Numerical Simulations. Materials; 2021; 14, 2033. [DOI: https://dx.doi.org/10.3390/ma14082033]

27. Meng, Q.; Yu, K.; Wang, R.; Qin, Y.; Wei, H.; Wang, X. Characteristics of Rocky Basin Structure of Yongshu Reef in the Southern South China Sea. Mar. Georesour. Geotechnol.; 2014; 32, pp. 307-315. [DOI: https://dx.doi.org/10.1080/1064119X.2013.764553]

28. Gourlay, M. Wave Set-up on Coral Reefs. 1. Set-up and Wave-Generated Flow on an Idealised Two Dimensional Horizontal Reef. Coast. Eng.; 1996; 27, pp. 161-193. [DOI: https://dx.doi.org/10.1016/0378-3839(96)00008-7]