Abstract
Focusing water waves is a potential technology improving the power generation of wave energy converters. Two semi-ellipsoidal reflectors, including long-axis opening and short-axis opening, were adopted to investigate the water-wave focusing effects. A 3D numerical wave tank was built and solved using the computational fluid dynamics (CFD) method. First, the wave fields around the reflectors against different wave periods were calculated. Furthermore, the wave elevation at monitoring locations of the reflectors against different wave steepness was investigated. Results demonstrate that the number of focusing points of the long-axis opening is more than that of the short-axis opening for short wave periods. The locations of focusing points move with the change of wave periods. However, for long wave periods, the waves can be focused over a large area in front of the reflector. The wave height at the focusing area overall becomes smaller with the increase in the wave periods. Additionally, the wave steepness has insignificant effects on the dimensionless wave height.
1 Introduction
The propagation of water waves is accompanied by huge energy, including kinetic energy and potential energy. The magnitude of water wave energy is proportional to the square of wave amplitude. Recently, the topic of wave concentration has attracted the interest of scholars in the field of marine engineering to improve the efficiency of power generation devices and reduce the cost. Water waves, like physical waves such as electromagnetic waves, acoustic waves, and elastic waves, have the characteristics of refraction, reflection, diffraction, and resonance. Based on the above wave characteristics, various wave concentration technologies have been proposed.
The lens is a typical wave energy gathering device, which uses refraction to focus the wave on a point in space. For deep water, an underwater submerged plate shaped as a convex lens was proposed to focus waves (; ; ). Then, the nonlinear water-wave focus of a submerged Fresnel lens has been investigated (). To increase wave transmission, the submerged plate was replaced by a horizontal cylinder array (; ). In shallow water, elliptical and bi-convex bathymetries were considered as the lens (). The conformal transformation technology was adopted to design a broadband gradient index lens using gradient water depth (). Recently, inspired by the photonic crystals, an array of vertical bottom-mounted cylinders was adopted to form a refraction-based focusing lens for long water waves (). Besides, metamaterial concepts have been adopted to design an epsilon-near-zero water waves focusing lens, in which the lens is a submerged plate shaped as a semi-circular (). Moreover, wave heights can be amplified at resonance conditions.
Moonpool is known as an operational hole of a drilling platform, where resonances can be excited at certain wave frequencies. proposed a moonpool platform–wave energy buoy (MPWEC). Numerical results validated that the wave energy conversion efficiency was significantly increased due to the resonance phenomenon inside the moonpool. Then, Lin et al. () experimentally studied the MPWEC introduced in with PTO systems. It was found that the moonpool can enhance the wave energy conversion in a certain range of wave frequencies. However, the moonpool will hinder the motion of the float buoy at some wave period. Cheng () et al. proposed a quadrate moonpool-type floating breakwater combined with an array of wave energy converters. By optimizing the size and the configuration of the multi-purpose platform, the floating system makes the concentration of wave energy over a wider frequency range.
Generally, the lens is sensitive to water depth, while the moonpool only works at the resonance frequencies. Compared with the above wave concentration methods, reflectors have attracted more interest in engineering due to their having fewer limitations. Besides, reflectors can be integrated with a breakwater to reduce cost.
Different shapes of water-wave reflectors have been considered. Straight reflectors are the most basic reflectors, which can increase the wave height twice by the vertical wall, known as standing waves. investigated the hydrodynamic performance of an array of truncated cylinders in front of the straight reflector. The heave exciting force of the cylinders can be improved at certain wave frequencies due to the reflected waves. An array of heaving Oscillating Buoy Wave Energy Converters (OB-WECs) attached at the weather side of a fixed straight breakwater has been numerically and experimentally studied by Zhao et al (). The results show that the existence of the straight breakwater can amplify the energy conversion performance of the WEC array. , Konispoliatis and Mavrakos (; ), Konispoliatis et al (; ), Kara (; ) arranged an array of truncated cylinders and an oscillating water column in front of a vertical breakwater, respectively. It was found that compared to the isolated converter array, the power efficiency of the converters in front of a breakwater is amplified at specific frequency ranges. investigated the cost-effective configuration of an array of WECs positioned in front of a vertical seawall in irregular waves. It was found that economically efficient wave power extraction can be achieved by installing multiple WECs in front of a straight reflecting wall, even in nearshore shallow water regions. Besides, the orthogonal reflection boundary has also been adopted to improve wave energy wave power extraction (). More than four times the heaving motion of the isolated WEC can be reached. It is known that a parabolic boundary can focus wave energy at a point. Thus, parabolic reflectors (; ; ; ; ; ; ) have been introduced to concentrate water-wave energy. Recently, hybrid systems containing a parabolic breakwater and wave energy converter (WEC) have attracted the research interests. Floating buoys (; ) and oscillating water column (; ; ) have been arranged in front of a parabolic breakwater to improve energy harvesting, respectively.
Above all, the curved reflector has great potential to reduce the cost of wave energy generation. To further expand the selection of reflectors, this study introduced an elliptical reflector and investigated the reflection behaviors against various wave periods and wave steepness. First, a three-dimensional numerical wave tank was developed using Computational Fluid Dynamics (CFD) to investigate the wave-focusing effects of elliptical energy-concentrating devices. Then, two types of semi-ellipse that the long axis and the short axis of the ellipse are used as openings have been considered, respectively. The wave focus against various wave periods was investigated. Finally, the effects of wave steepness on the wave focus by the elliptical reflector have been studied.
2 Theory
2.1 Governing equations
The water-wave focusing in this study is a wave-structure interaction problem; thus, a 3D numerical wave tank was established based on the Reynolds-Averaged Navier-Stokes (RANS) equations. Assuming the fluid is incompressible, the governing equations consist of the continuity equation and the Navier-Stokes (N-S) equations, which can be expressed as follows:
where Equation 1 represents the continuity equation, and Equations 2–4) represent the governing differential equations for the motion of viscous fluids. Equation 5 represents the time-averaged N-S equations. In these equations: u is the velocity component in the x-direction, v is the velocity component in the y-direction, w is the velocity component in the z-direction, ρ is fluid density, p is fluid pressure, τ is shear stress, f is the body force acting on the fluid element.
2.2 Free surface tracking method
In this study, the Volume of Fluid (VOF) method is used to capture the variations of the free surface. This method allows for the analysis of the position and shape of the multiphase flow interface, and it can predict the distribution and movement of the interface. The distribution of phases and the position of the interface are described by the phase volume fraction. Additionally, the numerical model applies the High-Resolution Interface Capturing (HRIC) technique to improve the accuracy of capturing the free surface variations. Equations 6, 7 represents the calculation method of phase volume fraction.
where Vi is the volume of phase i in the grid cell; V is the total volume of the grid cell; N is the total number of phases in the grid cell. In this study, N = 2.
3 Validation of the numerical method
The numerical wave tank used in this study is illustrated in Figure 1. The entire computational domain was defined with velocity inlet boundaries, a pressure outlet at the top, and a wall boundary at the bottom. The length of the wave tank is l = 72m, while the width is b = 48m. The water depth is 3m. Wave-damping zones were implemented around the periphery. Within the internal region, the three-dimensional Navier–Stokes equations (N–S equations) were solved directly. In contrast, force-based damping was applied in the external regions (damping zones) to guide the solution of the discrete N–S equations toward the theoretical solution within a prescribed distance. This approach not only reduces computational cost by allowing smaller domain sizes but also effectively suppresses wave reflection at the boundaries. The damping zones that are 1.0 times the wavelength at the inlet, 2.0 times the wavelength at the outlet, and 0.5 times the wavelength at the two sides were adopted.
Figure 1
3.1 Mesh convergence verification
The study focuses on regular waves with a wave height H = 0.07 m and a period of T = 1.5 s. Three mesh sizes were selected for the simulation, with specific mesh sizes and numbers shown in Table 1. The mesh sizes for Mesh A, Mesh B, and Mesh C are 12.5% of the base size in the x-direction (wavelength direction) and 1.56% of the base size in the z-direction (wave height direction) at the free surface. The wave gauge is placed at a distance of three times the wavelength from the wave generation boundary. Figure 2 presents the time history of the wave surface measured by the wave gauge under the three different mesh sizes. As shown in Figure 2, at three times the wavelength from the wave generation boundary, the relative error between the wave height measured by the wave gauge and the incident wave height is 6.98% for Mesh A, while the relative errors for Mesh B and Mesh C are both less than 5%. Considering both numerical accuracy and computational cost, Mesh B is chosen for the subsequent studies.
Table 1
| Mesh | Base size (m) | L/Δx | H/Δz |
|---|---|---|---|
| A | 0.34 | 82 | 13 |
| B | 0.28 | 100 | 16 |
| C | 0.22 | 127 | 20 |
The setup of cell number of mesh, and time-step for the convergence study.
Figure 2
3.2 Time step convergence verification
Next, the time step convergence of the numerical model with Mesh B size was verified, while the wave conditions were unchanged. Figure 3 shows the time history of the wave surface measured by the wave gauge for three different time step sizes. From the figure, it can be seen that as the time step size decreases, the curves approach the target wave shape. The simulations with time steps Δt = T/150 = 0.01 s, and Δt = T/300 = 0.005 s, produce nearly identical wave results. Considering that the simulation time for Δt = T/150 = 0.01 s is significantly shorter than that for Δt = T/300 = 0.005 s, a time step of Δt = T/150 = 0.01 s is chosen for the subsequent simulations.
Figure 3
4 Results and discussion
An ellipse with the equation in Equation 8 has been considered. Wave-focusing effects of two semi-ellipses that the long axis and the short axis of the ellipse are set as openings have been investigated, respectively, as shown in Figure 4. For each reflector, the wave elevation at five monitoring locations has been recorded, as shown in Figure 4. For the long-axis opening, five monitoring locations including (0m, 0m), (0.5m, 0m), (-0.5m, 0m), (0m, 0.45m), (0m, -0.5m) are selected, while five locations including (0m, 0m), (0.4m, 0m), (-0.4m, 0m), (0m, 0.4m), (0m, -0.4m) are selected for the short-axis opening. The incident waves propagate from left to right.
Figure 4
4.1 Wave-focusing effects against various wave periods
First, five wave periods, including T = 0.5s, 1.0s, 1.5s, 2.0s, and 3.0s, were adopted to investigate the wave-focusing effects in front of the elliptical reflectors. The wave steepness is set as WS = 0.02.
4.1.1 Time history of the wave distribution
The time history of the wave distributions in a period including t = 0T, T/4, T/2, 3T/4 of various wave periods (T = 0.5s, 1.0s, 1.5s, 2.0s, 3.0s) is shown in Figures 5–9, respectively. In each figure, the left column shows the results of the long-axis opening, while the right column shows the results of the short-axis opening.
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 5 shows the wave distribution of T = 0.5s. It can be found that the wave-focusing points (A/A0 > 1.0) exist both for the long-axis reflector and the short-axis reflector due to the interaction between the incident waves and the reflected waves. The focusing points are located at two areas, including the wall surface and the space in front of the wall. The number of the focusing points of the long-axis reflector is obviously larger than that of the short-axis. If a wave energy converter is arranged at the focusing point, more power can be generated. Moreover, the wave height in the lee-side of the reflector is significantly weakened due to the shielding effects of the wall, which can protect the structure in the lee-side.
Figure 6 demonstrates the results of T = 1.0s. Compared with that in Figure 5, it can be found that the number of focusing points reduces. The focusing points are also located on the wall surface and space in front of the wall. However, it should be noted that the location of the focusing points moves compared to that of T = 0.5s. The movement of the focusing points is not conducive to the capture of wave energy at fixed positions. Besides, the wave elevation at the lee side of the reflector is also weakened.
In Figures 7-9, the periods of the incident waves are T = 1.5s, 2.0s, and 3.0s, respectively, which means longer waves. It can be found that regional focus, instead of point focusing of the water waves, occurs for longer waves. Thus, the focusing area of long waves is larger than short waves. That means the movement of the focusing point does not need to be considered. Therefore, there are more options for the locations of the wave energy converter. Noted that the wave amplitude on the lee-side of the reflector is not significantly reduced, due to the diffraction of the long waves, especially for T = 3.0s, shown in Figure 9.
4.1.2 Wave heights at monitoring locations
To further investigate the wave focusing of the reflectors, the wave height at the monitoring locations shown in Figure 4 is calculated. The dimensionless wave heights (H/H0) at (0m, 0m), (0.5m, 0m), (-0.5m, 0m), (0m, 0.45m) of the long-axis opening are shown in Figure 10, while the dimensionless wave heights (H/H0) at (0m, 0m), (0.4m, 0m), (-0.4m, 0m), (0m, 0.4m) of the short-axis opening are shown in Figure 11.
Figure 10
Figure 11
In Figure 10a, it can be found that the wave height at (0m, 0m) is H/H0 = 1.72, which means the wave energy is focused. However, the wave heights at other locations are not amplified or even become smaller. As shown in Figure 10b, the wave height at (0m, 0m) is still larger than 1.0. Besides, the wave heights at (-0.5m, 0m) and (0.45m, 0m) are larger than 2.0, which means the focusing points move with T = 0.5s is change to T = 1.0s. In Figure 10c, it can be found than the wave heights at the locations inside the reflector ((0m, 0m), (0.5m, 0m), (0m, 0.45m)) are all larger than 1.0, while the wave height at the location outside the reflector ((-0.5m, 0m)) is smaller than 1.0. This means regional focusing inside the reflector happens for T = 1.5s. Seeing Figures 10d, e, the wave heights at all locations are larger than 1.0, which means the area of the regional focusing becomes larger for T = 2.0s and 3.0s (long waves). However, the value of the wave height overall becomes smaller with the increase of the wave period T. Noted that the height at (0m, 0m) is always larger than 1.0 for different wave periods (T = 0.5s, 1.0s, 1.5s, 2.0s, 3.0s), thus, the wave energy converter can be arranged at this location.
In Figures 11a, b, it can be seen that the wave heights at most locations are smaller than 1.0. Besides, the wave heights at one location are quite different for different wave periods T, which means the focusing points move with the change of T. Comparing Figure 10c with Figure 11c, it can be found that the wave heights at (0m, 0.4m) and (0m, 0m) of the short-axis reflector in Figure 11c is smaller than that of the long-axis reflector in Figure 10c, which means the focusing area of the short-axis reflector is smaller than that of the long-axis against T = 1.5s. As shown in Figures 11d, e, the wave heights at all locations are larger than 1.0, which means reginal focusing. Similar to long-axis opening, the value of the wave height overall becomes smaller with the increase of the wave period T.
4.2 Wave-focusing effects against different wave steepness
Three kinds of wave steepness (WS = H0/λ), including WS1 = 0.02, WS2 = 0.04, WS3 = 0.06, were adopted to investigate the nonlinear effects of the wave-focusing. Here, the wave period is set as T = 1.5s.
4.2.1 Long-axis reflector
As shown in Figures 12a, b, d, the dimensionless wave elevations (η/A0) at (0m, 0.5m), (0m, 0m), and (0.45m, 0m) are larger than 1.0, which means the water waves are focused at these locations. It can be found that waves are almost linear at these locations for different wave steepnesses. Due to that, there is no wave breaking, and the wave steepness has insignificant effects on the wave height. In Figure 12c, the wave elevation at (0.4m, 0m) is smaller than 1.0. An obvious nonlinear phenomenon can be found even for WS1 = 0.02, which is caused by the wave-structure interaction. A pair of small peaks and troughs appears. With the increase in wave steepness, the nonlinear phenomenon is more obvious. The wave peak becomes smaller due to wave breaking for larger WS; however, the wave trough also becomes smaller. Thus, the wave height has almost no change.
Figure 12
4.2.2 Short-axis reflector
Figure 13 demonstrates the dimensionless wave elevations (η/A0) at (0m, 0.4m), (0m, 0m), (-0.4m, 0m), and (0.4m, 0m), respectively. Significant nonlinear phenomena can be found at (0m, 0.4m) and (0m, 0m) as shown in Figures 13a, b, where the dimensionless wave height is smaller than 1.0. A pair of small peaks and troughs can be found. This means the wave nonlinearity leads to a significant reduction in wave height compared to the incident wave. In Figure 13c, the dimensionless wave height at (-0.4m, 0m) is close to 1.0, in which the wave steepness has little influence. The dimensionless wave height at (0.4m, 0m) is larger than 1.0, as shown in Figure 13d. The wave height has no obvious change with the increment of WS, due to no new wave peaks and wave through being generated.
Figure 13
5 Conclusion
A curved reflector can focus water waves, which is a potential technology in improving the efficiency of wave energy capture. In this study, two semi-ellipses with long-axis opening and short-axis opening were proposed, respectively. A 3D numerical wave tank is built based on the Navier–Stokes equations to solve this wave-structure interaction problem. The wave-focusing effects of these two reflectors against different wave periods (T) and different wave steepness (WS) were investigated, respectively. Results show that the types of wave focusing are different for different wave periods. Besides, wave nonlinearity has insignificant effects on the wave amplitudes in the focusing area. The main conclusions are as follows:
The wave distributions (A/A0) around the elliptical reflectors against different wave periods including T = 0.5s, 1.0s, 1.5s, 2.0s, 3.0s were calculated. For small-period waves, the number of focusing points of the long-axis opening is more than that of the short-axis opening. It should be noted that the locations of focusing points move with the change of wave periods. For long wave periods, regional focusing instead of point focusing of the water waves occurs. Besides, the wave amplitudes at the lee-side of the reflector are reduced for small wave periods but not significantly reduced for long-period waves.
The wave height (H/H0) at the monitoring locations of the long-axis opening and the short-axis opening was investigated, respectively. The results demonstrate that the wave height at one location changes with the change of wave periods in the range of small wave periods, which is not conducive to the capture of wave energy at fixed positions. For larger wave periods, the dimensionless wave heights at different locations are larger than 1.0, which confirms the regional focusing. However, the values of the wave height at different locations overall become smaller with the increasement of the wave period T.
The wave elevation (η/A0) at monitoring locations of the long-axis opening and the short-axis opening against different wave steepness, including WS1 = 0.02, WS2 = 0.04, WS3 = 0.06, was calculated to investigate the nonlinear effects. It was found that the wave steepness has insignificant effects on the dimensionless wave elevation (η/A0) if the dimensionless wave height is larger than 1.0 (H/H0 > 1.0). The small wave height (H/H0 < 1.0) at monitoring locations is mainly due to the generation of small peaks and troughs caused by wave nonlinearity.
The proposed semi-ellipses can be used to improve the power generation of a wave energy converter. In the future, the actual energy capture of the wave energy conversion device in front of the reflector will be further studied.
Nomenclature
U, Velocity vector (m/s); u, Velocity component along x-axis (m/s); v, Velocity component along y-axis (m/s); w, Velocity component along z-axis (m/s); ρ, Fluid density (kg/m3); p, Fluid pressure (Pa); f, body force (N); τ, Shear stress (N/m2); αi, Phase volume fraction; Vi, Volume of phase i in the grid cell (m3); l, Length of the numerical wave tank (m); b, Width of the numerical wave tank (m); Δx, Grid spacing in x-direction (m); Δz, Grid spacing in z-direction (m); Δt, Time step (s); η, Wave elevation (m); x, Horizontal position along x-axis (m); y, Horizontal position along y-axis (m); z, Vertical position along z-axis (m); t, Time (s); A, Wave amplitude (m); A0, Amplitude of incident waves (m); λ, Wavelength (m); H, Wave height (m); H0, Height of incident waves (m); T, Wave period (s); CFD, Computational Fluid Dynamics; RANS, Reynolds-Averaged Navier-Stokes; NS, Navier-Stokes; VOF, Volume of Fluid; HRIC, High Resolution Interface Capturing; WS, Wave steepness.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding author.
Author contributions
YW: Validation, Data curation, Visualization, Formal analysis, Methodology, Conceptualization, Software, Writing – original draft, Writing – review & editing, Investigation.
Funding
The author(s) declared that financial support was not received for this work and/or its publication.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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References
1
BobinskiT.EddiA.PetitjeansP.MaureA.PagneuxV. (2015). Experimental demonstration of epsilon-near-zero water waves focusing. Appl. Phys. Lett.107, 014101. doi: 10.1063/1.4926362
2
ChengY.XiC.DaiS.JiC.CocardM.YuanZ.et al. (2021). Performance characteristics and parametric analysis of a novel multi-purpose platform combining a moonpool-type floating breakwater and an array of wave energy converters. Appl. Energy292, 116888. doi: 10.1016/j.apenergy.2021.116888
3
ErtekinR. C.MonopolisG. M. (1986). “ Investigation of wave focusing over a parabolic step,” in Hydrodynamics of Ocean Wave-Energy Utilization. Eds. EvansD. V.de FalcãoA. F. O. (Verlag Berlin, Heidelberg: International Union of Theoretical and Applied Mechanics, Springer).
4
GriffithsL. S.PorterR. (2012). Focusing of surface waves by variable bathymetry. Appl. Ocean Res.34, 150–163. doi: 10.1016/j.apor.2011.08.004
5
HuX.ChanC. T. (2005). Refraction of water waves by periodic cylinder arrays. Phys. Rev. Lett.95, 154501. doi: 10.1103/PhysRevLett.95.154501
6
KaraF. (2021). Hydrodynamic performances of wave energy converter arrays in front of a vertical wall. Ocean Eng.235, 109459. doi: 10.1016/j.oceaneng.2021.109459
7
KaraF. (2022). Effects of a vertical wall on wave power absorption with wave energy converters arrays. Renew. Energy196, 812–823. doi: 10.1016/j.renene.2022.07.046
8
KongF.LiuH.SuW.AoJ.ChenH. (2019). Analytical and numerical analysis of the dynamics of a moonpool platform-wave energy buoy (MP-WEB). Energies12, 4083. doi: 10.3390/en12214083
9
KonispoliatisD. N. (2020). Performance of an array of oscillating water column devices in front of a fixed vertical breakwater. J. Mar. Sci. Eng.8, 912. doi: 10.3390/jmse8110912
10
KonispoliatisD. N.ChatjigeorgiouI. K.MavrakosS. A. (2020a). Near trapped wave phenomena in an array of truncated cylinders in a perpendicular arrangement in front of a vertical breakwater. Appl. Math. Model.83, 497–525. doi: 10.1016/j.apm.2020.03.005
11
KonispoliatisD. N.MavrakosS. A. (2020a). Theoretical performance investigation of a vertical cylindrical oscillating water column device in front of a vertical breakwater. J. Ocean Eng. Mar. Energy6, 1–13. doi: 10.1007/s40722-019-00147-6
12
KonispoliatisD. N.MavrakosS. A. (2020b). Wave power absorption by arrays of wave energy converters in front of a vertical breakwater: A theoretical study. Energies13, 1985. doi: 10.3390/en13081985
13
KonispoliatisD. N.MavrakosS. A. (2021). Hydrodynamic efficiency of a wave energy converter in front of an orthogonal breakwater. J. Mar. Sci. Eng.9, 94. doi: 10.3390/jmse9010094
14
KonispoliatisD. N.MavrakosS. A.KatsaounisG. M. (2020b). Theoretical evaluation of the hydrodynamic characteristics of arrays of vertical axisymmetric floaters of arbitrary shape in front of a vertical breakwater. J. Mar. Sci. Eng.8, 62. doi: 10.3390/jmse8010062
15
KudoK.TsuzukiT.ImaiK.AkiyamaY. (1986). Study on wave focusing by a horizontally submerged plate. J. Soc. Naval Architects Japan160, 217–225. doi: 10.2534/jjasnaoe1968.1986.160_217
16
LeeJ. W.CheungK. F. (1994). Effect of wave focusing structures in combined waves and a current. J. Korean Port Res.8, 67–78.
17
LiuH.YanF.JingF.AoJ.HanZ.KongF. (2020). Numerical and experimental investigation on a moonpool-buoy wave energy converter. Energies13, 2364. doi: 10.3390/en13092364
18
LoukogeorgakiE.ChatjigeorgiouI. K. (2019). Hydrodynamic performance of an array of truncated cylinders in front of a vertical wall. Ocean Eng.189, 106407. doi: 10.1016/j.oceaneng.2019.106407
19
MayonR.NingD.SunY.DingZ.WangR.ZhouY. (2023). Experimental investigation on a novel and hyper-efficient oscillating water column wave energy converter coupled with a parabolic breakwater. Coast. Eng.185, 104360. doi: 10.1016/j.coastaleng.2023.104360
20
MayonR.NingD.XuJ.FuL. (2024). Oscillating water column wave energy converter arrays coupled with a parabolic-wall energy concentrator in regular and irregular wave conditions. Coast. Eng.192, 104559. doi: 10.1016/j.coastaleng.2024.104559
21
MayonR.NingD. Z.ZhangC.ChenL.WangR. (2021). Wave energy capture by an omnidirectional point sink oscillating water column system. Appl. Energy304, 117795. doi: 10.1016/j.apenergy.2021.117795
22
MehlumE.StamnesJ. (1978). On the focusing of ocean swells and its significance in power production. Cent. Inst. Indust. Res. Blindern Oslo SI Rep.77, 1–38.
23
MehlumE.StamnesJ. (1979). “ Power production based on focusing of ocean swells,” in Proceeding of the First Symposium Wave Energy Utilization. (Gothenburg: Chalmers Univ. of Tech.) 29–35.
24
MurashigeS.KinoshitaT. (1990). Fundamental studies on ocean wave focusing. J. Soc. Naval Architects Japan168, 183–192. doi: 10.2534/jjasnaoe1968.1990.168_183
25
MurashigeS.KinoshitaT. (1992). An ideal ocean wave focusing lens and its shape. Appl. ocean Res.14, 275–290. doi: 10.1016/0141-1187(92)90032-F
26
NatarajanS. K.ChoI. H. (2024). Cost-effective optimization of an array of wave energy converters in front of a vertical seawall. Energies17, 128. doi: 10.3390/en17010128
27
RenJ.JinP.LiuY.ZangJ. (2021). Wave attenuation and focusing by a parabolic arc pontoon breakwater. Energy217, 119405. doi: 10.1016/j.energy.2020.119405
28
StamnesJ.LøvhaugenO.SpjelkavikB.MeiC.LoE.YueD. (1983). Nonlinear focusing of surface waves by a lens-theory and experiment. J. Fluid Mechanics135, 71–94. doi: 10.1017/S0022112083002967
29
van der WielR. J.KramerJ.van der VenP. P. D.BorsboomM.de JongM. (2016). “ Influence of a parabolic reflector wall on the sea state in an array of point absorbers wave energy converters,” in Progress in Renewable Energies Offshore: Proceedings of the 2nd International Conference on Renewable Energies Offshore. (London: Taylor & Francis Group) 1–9.
30
WangZ.ZhangP.NieX.ZhangY. (2015). Manipulating water wave propagation via gradient index media. Sci. Rep.5, 16846. doi: 10.1038/srep16846
31
XuJ.NingD.MayonR.ZhaoM. (2023). Hydrodynamic investigation of a parabolic breakwater for wave energy focusing. Phys. Fluids35, 097145. doi: 10.1063/5.0166601
32
YangC.ErtekinR. C. (1991). “ Numerical simulation of wave-energy focusing,” in OCEANS 91 Proceedings. (Honolulu, Hawaii: IEEE) 539–546.
33
ZhangC.NingD. (2019). Hydrodynamic study of a novel breakwater with parabolic openings for wave energy harvest. Ocean Eng.182, 540–551. doi: 10.1016/j.oceaneng.2019.04.056
34
ZhaoX. L.NingD. Z.LiangD. F. (2019). Experimental investigation on hydrodynamic performance of a breakwater-integrated WEC system. Ocean Eng.171, 25–32. doi: 10.1016/j.oceaneng.2018.10.036
35
ZhouB.WangY.ZhengZ.JinP.NingD. (2023a). Power generation and wave attenuation of a hybrid system involving a heaving cylindrical wave energy converter in front of a parabolic breakwater. Energy282, 128364. doi: 10.1016/j.energy.2023.128364
36
ZhouB.ZhengZ.ZhangQ.JinP.WangL.NingD. (2023b). Wave attenuation and amplification by an abreast pair of floating parabolic breakwaters. Energy271, 127077. doi: 10.1016/j.energy.2023.127077
Summary
Keywords
CFD, semi-ellipses reflectors, water-wave focusing, wave energy converter, wave period
Citation
Wang Y (2026) Water-waves focusing by an elliptical reflector. Front. Mar. Sci. 13:1752602. doi: 10.3389/fmars.2026.1752602
Received
23 November 2025
Revised
20 December 2025
Accepted
02 January 2026
Published
02 February 2026
Volume
13 - 2026
Edited by
Zhigang Zhang, Shandong University, China
Reviewed by
Rongquan Wang, Dalian University of Technology, China
Weijie Mo, Kyushu University, Japan
Shuang Liu, Ocean University of China, China
Updates
Copyright
© 2026 Wang.
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*Correspondence: Yihan Wang, wangyh_hrbeu@126.com
Disclaimer
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