The results of our analysis conducted so far indicate that the single-row terminator array of vertical cylinders shows a better energy extraction efficiency than that of the corresponding single-column attenuator array when the number of buoys is sufficiently large. Subsequently, we increase the number of rows (Nx) of this terminator array and investigate the energy extraction efficiency of the resulting arrays. Moreover, we use various dx or x-directional distance between adjacent rows in the same multi-row terminator array as the x-directional periodicity of the buoys reduces the energy extraction of this array. In particular, we study the effect of having various dx values within the same 4×8 terminator array of vertical cylinders on the energy extraction of this array under different configurations of it in detail.
3.4.1 Few rows terminator array
Generally, a multi-row terminator array of vertical cylinders can have all the already mentioned resonance phenomena occur in them. In addition, the x- and y-directional distances between adjacent rows and columns, respectively, of these arrays can result in different Bragg and Rayleigh resonance frequencies. Combining these distances also gives different Laue resonance frequencies in these arrays. The results of an analysis on the multi-row terminator arrays of vertical cylinders for
dy = 3a, 8a, and 14a are given in
Fig. 9. The green lines in this figure denote
kdx and
kdy divided by
π, respectively, for different combinations of array distances (
dy = dx,
dy = 2dx, and
dy = 0.5dx). The points of intersection between the green lines and the other blue horizontal lines, red vertical lines, and magenta curves satisfy the Bragg, Rayleigh, and Laue resonance phenomena, respectively. Subsequently, we increase the number of rows (
Nx) for a fixed number of columns (
Ny) of eight, as shown in
Fig. 10, and analyze the resulting arrays. Notably, these
Nx and
Ny values are sufficient to confirm the effect of multiple scattering of water waves by the buoys in these arrays.
Fig. 11 shows the non-dimensional average capture width of the multi-row terminator array of vertical cylinders for all the cases considered. In particular, the increment of the number of rows in these arrays reduces their energy absorption for a small
kdy, as shown in the left side graphs of this figure. This reduction is obvious when the maximum
kdy is smaller than the first Rayleigh resonance frequency.
The maximum energy absorption of general periodic WECs is limited by the normal component of the incident energy corresponding to the WECs’ maximum average capture width,(
Wa)
max = d, in the absence of oblique waves. Subsequently, we compute the maximum array gain
qmax for an array of
N rows in the absence of oblique waves (
Tokić and Yue, 2019) using the array gain given by
Eq. (11) and the maximum capture width of a heaving isolated buoy(
kW)
max = 1:
Fig. 2 of
Tokić and Yue (2019) shows that the array gain cannot always attain the maximum value. Therefore, the number of rows is crucial in the case of a small
kdy to achieve array gain. Hence, the increment of the number of rows in the multi-row terminator arrays of vertical cylinders results in a decrease of the average capture width of these arrays for small
kdy values.
This analysis also confirms the resonance effect in the multi-row terminator array of vertical cylinders for various combinations of
Nx and
Ny. The red, blue, and magenta dotted lines in
Fig. 11 denote the Bragg, Rayleigh, and Laue resonances, respectively, at each frequency. In addition, the gray line of this figure denotes the simultaneous existence of two resonance phenomena, and the black vertical line in this figure denotes the occurrence of all resonances simultaneously at this frequency. First, the effect of Bragg resonance seems small in all these array configurations. Next, the magnitude of the non-dimensional average capture width oscillates between resonance points generally. In addition, a lot of resonance points occur when
dx and
dy are sufficiently large. Particularly, several Laue resonance points are generated under this condition. Therefore, these arrays’ non-dimensional average capture width becomes extremely oscillating if the distance between adjacent buoys of these arrays is relatively large, as shown in the upper right graph of
Fig. 11. However, a large amplitude occurs only when the number of rows in these arrays is smaller than five. Further, the extraction efficiency of the multi-row terminator array of vertical cylinders decreases when the number of rows in it is larger than four generally. Consequently, a smaller value for the number of rows in these arrays gives better performance than a corresponding single-row terminator array, provided the former has a moderate
kdy.
3.4.2 A 4×8 multi-row array
Tokić (2016) attempted to find the local maximum of extracted power of a multi-row terminator array by using an optimization scheme for this array’s x- and y-directional distances between its rows and columns, respectively (
dx and
dy), that can vary inside the array. He found that the local maximum of this extracted power could be obtained for various
dx and constant
dy values within the array under regular water waves. Moreover, a regular array achieves a local maximum of extracted power at least when the inclined angle of irregular water waves is small (
Tokić and Yue, 2021).
Tokić and Yue (2021) also concluded that a perturbed position of each buoy results in less extracted power of this array.
Our analysis of the single-row terminator array of vertical cylinders shows that a constant
dy is constructive. In addition, achieving a local maximum of extracted power in a regular array under irregular water waves is possible only with an unconstrained total length of the array. On the other hand, a constrained overall length of an array together with various
dx values within it results in better energy extraction efficiency of the array under some conditions. So, we investigate the extracted power of a 4×8 multi-row array of vertical cylinders with various
dx values within it though under regular water waves. In addition, the distance between rows of this array is changed. In particular, we choose one regular 4×8 multi-row array of vertical cylinders and six variants of it with at least two same x-directional distances between the rows, as shown in
Fig. 12. In addition, four combinations of distances are chosen for the analysis (
dx = 3a, dy = 5a;
dx = 3a, dy = 12a;
dx = 6a, dy = 5a;
and dx = 6a, dy = 12a). The
ds in these arrangements is always greater than
dx.
The results of the analysis on the 4×8 multi-row arrays of vertical cylinders are summarized in
Fig. 13. The black line in this figure shows the non-dimensional average capture width of a
regular 4×8 multi-row array of vertical cylinders. Other colorful lines denote the results for the corresponding arrays with various
dx in them. In particular,
dx influences the relative magnitude of the non-dimensional average capture width between this regular array and the other corresponding arrays with various
dx. In addition, a small
dx is constructive only in these arrays with various
dx. Notably, various small
dx (2.8a <
dx < 3.5a) values result in larger energy extraction efficiency of any of these arrays with various
dx than that of a corresponding regular array with the same
dx. However, the details of the analyses giving this finding are not reported here. The graphs corresponding to
dx = 3a in
Fig. 13 have the colorful lines representing the non-dimensional average capture widths of these arrays with various
dx which are larger than that (represented by the black line in this figure) of the corresponding regular array. On the other hand, the black line in the graphs corresponding to
dx = 6a represents the non-dimensional average capture width of a regular 4×8 multi-row array of vertical cylinders, which is greater than those (represented by the colorful lines in this figure) of the corresponding arrays with various
dx values. The energy extraction efficiency of all these multi-row terminal arrays of vertical cylinders with various
dx is better than that of the corresponding regular array almost in all frequency ranges. So, we expect their similar high performances under irregular water waves. Unfortunately, it is difficult to find which of the 4×8 multi-row arrays of vertical cylinders with various
dx is better. However,
SSX and
XXS types of these arrays have relatively large and small efficiencies, respectively, in many cases.