1. Introduction
Renewable energy provides a reliable and sustainable alternative to traditional fossil fuels. Its development and application are regarded as a high priority in many countries based on environmental protection, national energy security, and critical technology advancement. The ocean is a renewable energy resource; the maximum annual potential capacity of ocean energy, including tidal currents, waves, and thermal gradients, can reach 170,400 TWh/year [
1]. According to the study conducted by Vining and Muetze [
2], wave-energy potential is identical to the total nuclear energy or hydropower. Among different ocean energy technologies, wave-energy converters (WECs), transforming the kinetic and potential energy of moving waves into electric energy, have undergone rapid development [
3]. When considering the transfer of these technologies to practical applications, the efficiency and stability of wave energy are always of major concern in feasibility studies. The application of WECs and the generated power strongly depend on wave conditions.
According to Airy wave theory [
4,
5], the average mechanical energy flux, including both kinetic and potential energy, per unit width over a wave period,
, can be written as
and the wave power,
, is defined as
where
is water density,
is gravitational acceleration,
is the wave height, and
is the group velocity of a wave. It is evident from the abovementioned equations that wave power is proportional to the square of wave height. Therefore, a suitable choice for the development of a wave farm generally requires sufficient wave heights. However, some coastal areas may not be suitable for the development of wave farms because of limited wave heights, along with other constraints. For instance, ocean waves in Taiwan are strongly dominated by monsoons, except for extreme typhoon events. Monsoons can be generally categorized into the southwest monsoon in summer and the northeast monsoon in winter [
6]. According to the statistics of the Central Weather Bureau [
6], significant wave heights are commonly less than 1 m for most seasons and may reach 5 m in winter, which is more suitable for wave-energy harvesting. However, the electricity peak-load periods are mostly in summer, owing to the large air conditioning demands [
7]. When developing wave farms for such demands, concentrating the wave energy and enhancing its density in low or moderate wave seasons would be an appropriate solution in coastal areas.
When waves are focused to a specific location, the wave heights increase within a specific area. The area can be considered for planning and designing wave farms. Waves refract, diffract, and shoal when they pass over submerged structures or changing bathymetries. Many studies have investigated induced wave deformation using mathematical theories, numerical simulations, and model experiments. Berkoff et al. [
8] developed an elliptical-type mild-slope equation based on Airy wave theory. They numerically investigated the phenomenon of refraction and diffraction on waves passing over an elliptical shoal by implementing a finite element method model. It was found that waves concentrate behind the shoal and the simulated results conformed with the experimental results. Kirby and Dalrymple [
9] simulated weakly nonlinear waves that satisfy the conditions of an Ursell number
Ur < O(1). The waves conforming to the parabolic-type equation based on Stokes expansion were propagated through an elliptical shoal, and the results were compared with experimental results reported in [
10]. Ebersole [
11] developed an efficient numerical model based on the finite difference method, solving both refraction and diffraction of waves propagating across irregular bottom configurations. The model was validated for both the semicircular shoal tests at the laboratory scale performed by [
12] and a practical application surrounding the Oregon Inlet in North Carolina. Griffiths and Porter [
13] also numerically investigated wave focusing of regular waves induced by variable bathymetry with a full three-dimensional (3D) linear wave theory approximated by the depth-averaged modified mild-slope equation and a Green’s function approach. The bathymetry profiles were selected based on the concept of optical refraction and contained both types of elliptical and biconvex lenses. The results suggested that an elliptical lens is more effective than a biconvex lens for wave refraction, but it requires a larger raised area. Weng et al. [
14] conducted experiments to explore the effects of wave-energy enhancement using a crescent-shaped shoal. The shoal was inspired by moon blocks, used as divination tools in Poe divination. It has a convex profile, facing the incident wave direction, with a positive decreasing slope and a concave profile with a negative increasing slope connected by the crest line. According to the experimental results, the increased wave energy is eight times higher than the incident wave energy, and the maximum ratio could reach up to fourteen times higher than the incident wave energy under the test conditions.
However, for practical applications of wave-energy harvesting, it is typically more desirable to have a controlled dimension of the submerged structure compared to the construction of a shoal. Mehlum [
15] resolved the problem of waves travelling across a submerged cylinder using certain recursive relations. The problem was studied in linearized wave theory and inspired by the phase shifts induced by cylinders that cause wave focusing. Stamnes et al. [
16] compared experimental results of waves propagating through a Fresnel-type wave lens conducted in a large outdoor basin with those of linear and nonlinear theoretical wave theories. The results indicated that the nonlinear model represented the experiments well and there were variations between the results of the linear model and the measured ones. However, the efficiency of wave focusing for wave-energy harvesting did not seem to be strongly diminished by nonlinear effects. Kudo et al. [
17] resolved wave deformation when waves pass through a submerged horizontal convex thin plate in deep water using a 3D linear numerical method and experiments. The phenomenon of wave focusing was identified and the maximum amplified wave heights were approximately three times higher compared to the incident wave heights at the edge of the plate. Tsuzuku et al. [
18] proposed a crescent-shaped plate composed of arcs of an ellipse and a circle based on the law of refraction, and both experimental and numerical studies were conducted. The design of the crescent-shaped plate was considered promising because the position of the induced wave focusing could be better controlled outside the plate compared to the convex plate. Murashige and Kinoshita [
19] derived a hydrodynamic singularity distribution for an ideal wave-focusing lens based on its slenderness and high incident wave frequency. The results suggested that its sectional shape should fulfill essential conditions equivalent to geometrical optics and a convex circular cylinder performs better for wave focusing using the slender ship theory. McIver and Urka [
20] investigated the properties of wave reflection and transmission of a circular arc plate submerged in deep water under the conditions of Airy wave theory. The results were compared with those of a circular cylinder to determine the feasibility of replacing a circular cylinder for applications of water wave lenses. Teigen [
21] attempted to investigate both focusing and defocusing effects of waves for the purposes of wave-energy harvesting and coastal protection. Wave propagation through four different simple horizontal plate shapes, including circular, rectangular, hyperbolic, and twin rectangular plates, was simulated with the linear version of WAMIT. Newman [
22] attempted to maximize the wave focusing caused by different submerged plate shapes with the combination of tools of PRAXIS for optimization and WAMIT for wave interaction with structures. The plate shapes were planforms defined by a Fourier series instead of simple geometrical shapes. The results showed that the maximum wave focusing lies between five and fifteen times higher than the incident wave height at one wavenumber.
In this study, we investigated the properties of wave focusing of a horizontal crescent-shaped plate based on the framework of the crescent-shaped shoal proposed in [
14] under regular wave conditions. When waves are focused to a specific location, owing to the reflection, refraction, diffraction, and shoaling, the wave heights, which represent the wave energy, increase; the wave energy density within a specific area is also enhanced. This area can be further considered for planning and designing wave farms and installing WECs to increase the overall efficiency of wave-energy harvesting. Among different types of WECs, an array of distributed point absorber WECs is considered to be potentially employable in the wave farm. Owing to this selection criteria, the directions of waves will not cause any application issues and higher wave heights would result in higher energy harvesting. The investigation was conducted both experimentally and numerically for separate directions of incident waves under different wave conditions. All experiments were performed in the wave basin at the Ocean Engineering Laboratory of the National Taiwan Ocean University, and the generated waves were controlled to be nonbreaking. A 3D numerical model implemented with the boundary element method (BEM) based on Airy wave theory was employed, and the calculated results were compared with the experimental ones. Planar wave-height distribution and cross-sectional wave-height elevation derived from both calculations and experiments were presented by the wave-height ratio
, a dimensionless measure of wave height relative to the incident wave height. The wave-focusing characteristics induced by the submerged crescent-shaped plate for different incident wave directions, regular wave conditions, and submerged depths are discussed in the hope of providing insightful information for further possible applications of ocean-wave-energy harvesting.
2. BEM-Based Numerical Model
The domain of interest is shown in
Figure 1, and a crescent-shaped, horizontal plate is submerged and installed in the center of the domain. The submerged plate shape is formed by two orthogonal ellipses with different axes with the same center point, which is detailed in
Section 3. A Cartesian coordinate system is chosen and the z-axis projects vertically upwards from its origin on the undisturbed free surface. The domain of interest is partitioned into Regions I and II (
Figure 1). It is assumed that waves enter Region I from infinity, and an auxiliary boundary is far from the plate, where reflected, refracted, and diffracted waves induced by the plate have no effect. Region II is defined by the auxiliary boundary and the submerged plate.
The fluid within the domain of interest is assumed to be ideal, and the fluid motion satisfies Airy wave theory. When a train of small amplitude waves travels into the domain from infinity, the motion of the fluid can be described in terms of the velocity potential,
, where
is amplitude,
is frequency,
is acceleration due to gravity, and the potential function
within the domain of interest satisfies the Laplace equation:
where
for different sub-regions, Regions I and II, separated by an auxiliary boundary
, are denoted as
and
.
2.1. Potential Formation in Region I
Region I is assumed to be of constant water depth
h and is bounded by the auxiliary boundary
and the boundary at infinity. The auxiliary boundary is assumed to be far from the plate, and the induced evanescent waves have no effect on it. The potential function
can be written in the form of depth- and spatial-dependent functions with the help of separation of variables:
where
is the incident wave number, the root of the linear dispersion relationship
, and
and
are potential functions of incident waves and scattered waves, including reflected and diffracted waves induced by the plate, individually. The potential function
can be described as
where incident waves bisect the x-axis at an angle
, as shown in
Figure 1.
The potential of scattered waves
within Region I satisfies the Helmholtz equation:
when substituting Equation (4) into Equation (3). Region I is enclosed by the auxiliary boundary and a boundary at infinity. When scattered waves
approach infinity, the boundary condition applied at infinity satisfies the Sommerfeld radiation condition:
By applying Green’s integral technique, the potential function of scattered waves,
, within Region I of constant water depth can be written as
where
is the potential function of scattered waves on the boundary represented by the local
-
coordinate system,
is the normal outward derivative on the boundary,
, and
is the 0th order Hankel function of the first kind. The constant
is defined to be 0.5 when points lie on a smooth part of boundary, and
= 1 for any point within the domain. Equation (8) can be further discretized on the boundary according to BEM and the matrix form is given by:
where [
G*] is a coefficient matrix related to the shape of the geometrical boundary [
23], and
represents the normal derivative
on the boundary.
2.2. Potential Formation in Region II
Region II is enclosed by the auxiliary boundary Γ
1, the free surface Γ
2, the surface of the submerged plate Γ
3, and the sea floor Γ
4. It is a closed three-dimensional domain. Similar to the approach described above applying Green’s integral technique, the potential function
within Region II can be expressed as
where
is a general potential function on the boundary represented by the local
-
-
coordinate system,
, and
= 1 when any point
is within the domain of Region II, and
= 0.5 when it is on the boundary. Similarly, Equation (10) indicates a matrix form when the boundaries enclosing Region II are discretized according to BEM as
where [
G] is a coefficient matrix related to the shape of the geometrical boundary, and
represents the normal derivative
on the boundary.
2.3. Potential Resolving
As described above, Region II is a three-dimensional domain enclosed by the auxiliary boundary Γ
1, the free surface Γ
2, the submerged plate surface Γ
3, and the sea floor Γ
4. To resolve the potentials within Region II, the following boundary conditions must be satisfied:
where Equation (12) describes the boundary condition at the free surface fulfilling both the dynamic and kinematic boundary conditions, and Equations (13) and (14) express the impermeable boundary condition at the submerged plate surface and the sea floor. In addition, continuity of mass and the energy flux must be satisfied along the auxiliary boundary:
After substituting Equation (4) into Equation (15), multiplying with
, and integrating along the water depth from
to
, the normal derivative of the potential function
on the auxiliary boundary can be described by
and
as
where
.
Likewise,
on the auxiliary boundary can be expressed in terms of
and
after substituting Equation (4) into Equation (16):
Finally, a matrix form describing the relationship between
and
on the auxiliary boundary can be derived by discretizing and substituting Equations (17) and (18) into Equation (9):
where
, and
and
are coefficient matrices defined in [
23].
To resolve the potentials within Region II, a system of equations can be derived by substituting boundary conditions, Equations (12)–(14), and (19), into Equation (11). When the boundaries are categorized into four different ones,
,
,
, and
, as described earlier, Equation (11) becomes:
where the subscripts
i and
j indicate the corresponding boundaries. After sorting, Equation (20) can be organized as
The potential function and its derivative, and , on the boundary of Region II can be determined first when Equation (21) is resolved. The potential of Region II can then be computed while solving Equation (11) for given as 1. Finally, the wave-height ratio of the free surface within Region II, a dimensionless wave height relative to the incident wave height representing the wave field throughout this study, can be defined and derived as , where describes the free surface amplitude and is defined as .
3. Experimental Setup
A physical model of the crescent-shaped plate, composed of two orthogonal ellipses with different axes, was built and placed in the wave basin at the Ocean Engineering Laboratory of the National Taiwan Ocean University. The geometric components of the plate are shown in
Figure 2. Two orthogonal ellipses, the outer ellipse (E
O) and the inner ellipse (E
I), overlap at the center, and the lines drawn from the intersections form an angle (θ) of 120°. The major and minor axes of E
O and E
I are 4.00 m and 2.28 m, and 3.10 m and 1.75 m, respectively; the thickness of the plate is 0.30 m (
Table 1).
The wave basin is approximately 50 m long, 50 m wide, and 1 m high, with a directional irregular wave generator. The wave generator is 28 m wide and composed of 56 sets of wave paddles. A 2 m wide artificial beach with a slope of 1:3 was installed around the basin for the purpose of wave-energy dissipation. The plate model was placed in the middle of the basin and the top end was 20.6 m from the wave generator. In addition, ten fixed supports were attached at the bottom of the plate to avoid vertical movements during experiments when waves passed.
Figure 3 shows the plate model in the basin before water was added. The experimental water depth was set to be 0.5 m and the plate was submerged 0.1 m below the water surface, which is 0.2 h. An incident wave gauge was placed along the centerline of the plate and 9 m from the wave generator. All experiments were conducted with nonbreaking waves and the regular wave conditions are listed in
Table 2. It should be noted that the dimensionless wave condition
was chosen, indicating different wave conditions. The greater the value of
, the shorter the wave period, and vice versa. Wave heights were measured by resistance wave gauges at 0.2 m intervals within an 8 × 8 m square region incorporating the plate area, and the center of the square region was located 1.375 m from the top end of the plate. As mentioned previously, the numerical results are presented by
values describing the steady state wave field. To compare between the unsteady state experimental results and the steady state numerical results, every measured time series of water surface elevation was converted to the corresponding wave height and
value.