3.1. General Remarks
The BIE-based numerical model described in
Section 2 is applied for an array of
identical, semi-immersed oblate spheroidal WECs (
Figure 1) with semi-major axis
m and non-dimensional semi-minor axis (draft)
. These dimensions have been selected based on [
28], who demonstrated the enhanced, as compared with other WEC geometries (i.e., cylinder, hemisphere, and prolate spheroid), absorption ability of a corresponding single isolated oblate spheroidal WEC at infinite water depths. All five WECs are considered to have the same PTO characteristics; thus,
,
. Moreover, the damping coefficient of the PTO is tuned to a constant value, so that maximum energy absorption is achieved at the natural frequency of a single isolated WEC (e.g., [
22,
29,
30,
31]). Accordingly and in line with [
32], the aforementioned
value is taken equal to the heave radiation damping of a single isolated WEC at its heave natural frequency,
, i.e.,
. On the basis of the hydrodynamic analysis of an isolated oblate spheroidal WEC,
has been calculated equal to
rad/s, resulting in
Nm/s.
Initially, relevant results are compared with those corresponding to arrays of semi-immersed cylindrical and hemisphere-shaped WECs (
Figure 2) in order to demonstrate the enhanced absorption ability of the aforementioned array. Aiming at comparing WECs that have the same
, irrespectively of their geometry, the geometrical characteristics of the cylindrical and the hemisphere-shaped WECs have been defined, so that
= 2.4 rad/s for both geometries, as in the case of the oblate spheroidal WECs. These characteristics along with the corresponding
values are shown in
Table 1, where the notations of
Figure 2 are taken into account. For both the cylindrical and the hemisphere-shaped WECs, the
values of
Table 1, equal to
, have been obtained from the hydrodynamic analysis of the corresponding isolated body. This, in turn, leads to the consideration of different
values among the examined geometries, due to the existence of different hydrodynamic properties (
Figure 3).
For each of the three examined arrays, the WECs are placed in front of the wall at a distance
of
m and they are distributed uniformly within the array with non-dimensional center-to-center spacing equal to
. Comparisons are made for two cases of non-dimensional length of the wall,
equal to
and 72 and
and
, respectively, with
denoting the distance between the centers of the two outer WECs in the array with the corresponding edges of the wall, as shown in
Figure 1a. It is noted that for a given WEC geometry, non-dimensional quantities are defined using the corresponding value of
, included in
Table 1.
Next, we investigate and assess the effect of the ratios
and
on the hydrodynamic behavior and the power absorption ability of the array of the five, semi-immersed oblate spheroidal WECs. For the first design parameter, six different values of
(
Table 2) are examined for
and
, and for the second examined design parameter, three different values of
are considered (
Table 2) for
and
. Results for the isolated array (i.e.,
) are also cited for illustrating more clearly the effect of the presence of the wall on the performance of the array. It is mentioned that the value of
has been chosen based on preliminary tests that revealed adequate energy absorption ability for this center-to-center spacing as compared with other
values examined.
In all the above cases, the array and the wall are considered to be situated in a liquid region with
m and they are subjected to regular incident waves propagating at angle
deg (
Figure 1a) with
varying between 0.01 and 4.0. The heave exciting forces are given normalized by
, while the symmetry of the array with respect to the examined incident wave direction leads to the same exiting forces and responses for WEC1 and WEC5 (outer WECs) and for WEC2 and WEC4 (
Figure 1a with
).
3.2. Comparison of Arrays with Different WEC Geometries
In
Figure 4, the heave exciting forces,
, applied on the semi-immersed oblate spheroidal WECs of the examined five-body array are compared with the corresponding ones applied on the WECs of the arrays that consist of semi-immersed cylindrical and hemisphere-shaped WECs (
Figure 2). The results of
Figure 4 correspond to the case of
, while analogous conclusions can be drawn for
. For all the oblate spheroidal WECs,
starts its variation from the limiting value of one (at
rad/s), as exists in the case of the isolated array (see for example Figure 7 in
Section 3.3), and accordingly it increases quite rapidly up to
rad/s, where the
peak (global maximum) occurs. A rapid decrease follows leading to a local minimum of
at
rad/s
rad/s, and for the remaining examined
values,
varies quite intensively, having multiple peaks and successive local minima, with successively decreasing values towards higher frequencies. The utilization of hemisphere-shaped WECs does not introduce significant differences in the values and the variation patterns of the heave exciting forces. However, it is worthwhile to note that for the specific WEC geometry the observed global maximum is shifted at a slightly larger
as compared with the oblate spheroidal WECs. This shift becomes more pronounced in the case of the cylindrical WECs; moreover,
applied on the cylindrical WECs varies more smoothly at
rad/s as compared with the oblate spheroidal and the hemisphere-shaped WECs. The above characteristics lead to different
values of the cylindrical WECs as compared with the other two examined geometries at specific
values (e.g., at
rad/s
rad/s for WEC2 and WEC4,
Figure 4b, at
rad/s
rad/s for WEC3,
Figure 4c).
Figure 5 shows the comparison of
between the three examined WEC geometries. Again, the results correspond to arrays placed in front of a wall with
, while similar conclusions can be drawn for the case of
. Irrespectively of the WECs’ geometry and the position of the WEC within the array,
follows, in general, the variation pattern of the heave exciting forces which is characterized by the occurrence of two distinctive peaks; one at
rad/s
rad/s as a result of the existence of the peaks in
in the concerned frequency range (
Figure 4) and a second peak at
rad/s
rad/s attributed to resonance phenomena. Moreover, a local minimum is observed at
rad/s
rad/s, in absolute accordance with the variation pattern of the heave exciting forces (
Figure 4). Except at
rad/s
rad/s, the deployment of hemisphere-shaped WECs within the array does not lead to substantial differences of
as compared with the case of the oblate spheroidal WECs. In the concerned frequency range, where resonance phenomena occur, the consideration of a smaller
value for the hemispheres (
Table 1) has a more pronounced effect on
, leading, therefore, to slightly larger values for
as compared with those of the oblate spheroids. In a similar manner, the utilization of an even smaller
value for the examined cylindrical WECs (
Table 1) leads to a significant increase of
at
rad/s
rad/s in the case of WEC
j,
and
as compared with the corresponding oblate spheroidal and the hemisphere-shaped WECs. The same holds for WEC3 (
Figure 5c) at
rad/s
rad/s, except at
rad/s, where
of WEC3 shows a local minimum in absolute accordance with the corresponding variation of the heave exciting force (
Figure 4c). Finally, for all cylindrical WECs, the existence of different
values at specific frequency ranges (e.g., at
rad/s
rad/s,
Figure 4) as compared with the rest examined WEC geometries leads also to different
values at these frequency ranges.
As far as the absorbed mean power is concerned,
Figure 6 shows the comparison of
among the three examined five-body arrays. For both
and
and for all three arrays, the variation pattern of
is characterized by the existence of a first (local) peak at
rad/s
rad/s and a second (global) peak at
rad/s
rad/s, in accordance with the variation of
, as previously discussed for the case of
. Moreover, local minima of
occur at specific
values as a result of the relevant substantial decrease of
. Compared to the cylindrical WECs, the deployment of either the hemisphere-shaped or the oblate spheroidal WECs greatly improves the power absorption ability of the array, since for the latter WEC geometries significantly larger
peak values occur, and, additionally, for
rad/s the frequency range, where adequate amount of power is absorbed, becomes wider. All the above are attributed to the consideration of a quite smaller
value in the case of the cylindrical WECs (
Table 1), as a result of their intrinsic hydrodynamic characteristics. Therefore, although this
value results in a significant increase of
at the frequency range where resonance phenomena occur (
Figure 5), it reduces remarkably the power absorption ability of the array. Following a similar rationale, the deployment of hemisphere-shaped WECs within the array leads to slightly smaller
peak values as compared with the oblate spheroidal WECs. Consequently, the oblate spheroidal WECs can be considered to have the best power absorption ability among all the three examined WEC geometries.
Finally, it is worthwhile to mention that for a given WEC’s geometry, the length of the wall affects at a small degree
. Specifically, in the case of
(
Figure 6a), the existence of more intense wall-edge effects reduces the power absorption ability of the array at
rad/s
rad/s, leading to slightly smaller
peak values in this frequency range as compared with
(
Figure 6b). However, at smaller frequencies, wall-edge effects seem to have a positive effect on
, since at
rad/s
rad/s slightly larger
values are observed for
as compared with
.
3.3. Effect of WECs’ Distance from the Wall on the Array’s Performance
The effect of the distance of the semi-immersed oblate spheroidal WECs from the wall on the WECs’ heave exciting forces is shown in
Figure 7, where the variation of
for all WECs in the array as a function of
is presented for all
values examined, as well as for the isolated array. For WEC
j,
, and
(
Figure 7a,b) and for
, the variation of
begins from the limiting value of
at
rad/s and it is characterized by a rapid increase up to
rad/s, where the peak of
(global maximum) occurs. Accordingly,
decreases quite rapidly up to
rad/s, where it obtains its first local minimum, which approaches zero. Finally,
is increased resulting in a second peak at
rad/s. Analogous variation pattern is observed for all the examined
values. However, by successively increasing the array’s distance from the wall, the first local minimum of
and, thus, its second peak are successively shifted at smaller frequencies, while the values of the latter peaks are also consecutively increased. The above trends introduce differences on the values of
at specific frequency ranges. For example, the increase of
leads to larger
values at
rad/s
rad/s, while the opposite holds true at
rad/s
rad/s. Analogous conclusions can be drawn for the heave exciting forces applied on the middle WEC of the array (
Figure 7c). However,
of WEC3 obtains one more distinctive peak at
rad/s for the two smallest examined WECs’ distances from the wall (i.e.,
and
).
The results of
Figure 7 demonstrate clearly that the presence of the wall, irrespectively of the distance of the WECs from this boundary, affects significantly the heave exciting forces of all the WECs in the entire examined frequency range, since for all
values,
does not show the continuous smooth decrease as in the case of the isolated array. Compared to the latter array, the existence of the wall boundary in the leeward side of the WECs at small distances from them (i.e.,
and
) leads also to a substantial increase of
at
rad/s
rad/s and at very high frequencies (
rad/s
rad/s), while the placement of the array at further distances from the wall (i.e.,
,
,
, and
) results in a significant increase of
at
rad/s
rad/s, as well as at
rad/s
rad/s, where resonance phenomena of the bodies are anticipated. From a physical point of view, the above trends can be related to the formation of characteristic patterns of the diffracted wave field around the WECs, as commended in
Section 3.5.
Figure 8 shows the effect of
on
for all the oblate spheroidal WECs. In this figure, results corresponding to the isolated array are also included to demonstrate more clearly the effect of the presence of the wall on
. In the case of
and for WEC
j,
, and
(
Figure 8a,b), the variation pattern of
is quite intense and it is characterized by three distinctive peaks. The first one (global maximum) and the third one occur at
rad/s and
rad/s, respectively, as a result of the relevant
maximum values (
Figure 7), while the second peak is observed at
rad/s and it is related to resonance phenomena. Moreover, a characteristic local minimum, following the preceding rapid decrease of
, occurs at
rad/s, in absolute accordance with the variation of the heave exciting forces (
Figure 7). A similar variation pattern is observed for all examined
values. However, by successively increasing
up to
, the aforementioned local minimum and resonance-related peak of
are successively shifted at slightly smaller frequencies as compared with the case of
, while the values of the latter peaks are increased substantially. Same conclusions can be drawn when
is further increased up to
; nevertheless, the placement of the WECs at one of the two largest examined distances from the wall does not introduce any significant differences in the peak values of
at
rad/s
rad/s as compared with
. It is also worthwhile to note that at
rad/s
rad/s the increase of
leads to a smooth reduction of
values, while only the curve of
has a distinctive third
peak at very high frequencies (i.e., at
) as in the case of
. All the above are in absolute accordance with the results of
Figure 7. Finally, with regard to WEC3, which is situated in the middle of the array (
Figure 8c), conclusions similar to the case of WEC
j,
, and
can be drawn. However, for all
values examined,
obtains one more distinctive peak at
rad/s.
Compared to the isolated array, the results of
Figure 8 illustrate that the presence of the wall, has a significant impact on
of all the WECs in almost the entire examined frequency range. Specifically, on the one hand, when the array is placed in front of the wall at a small distance from the wall (i.e.,
and
), the WECs’ heave responses increase significantly mainly at
rad/s
rad/s, while a great reduction of
occurs at
rad/s
rad/s, especially for
. On the other hand, by increasing
to values larger than 2,
, increases at
rad/s
rad/s, but, most importantly in the frequency range, where resonance phenomena occur.
Finally, the effect of
on the power absorbed by the whole array is shown in
Figure 9, where
is plotted as a function of
for all
values examined, as well as for the case of the isolated array. For
, a significant amount of power is absorbed at
rad/s
rad/s, where
obtains its maximum value approximately equal to
kW/m
2. Moreover, additional
peaks with smaller values occur at
rad/s and
rad/s as a result of the relevant
peak values (
Figure 8). By increasing
to
, a significant amount of power is absorbed at larger wave frequencies and, more specifically, at
rad/s
rad/s, where resonance phenomena also occur. Moreover, the power absorption ability of the array is substantially enhanced, since the maximum value of
, occurring at
rad/s, is approximately equal to
kW/m
2. Similar behavior is observed for the rest of the examined
values. However, the increase of
from 2.0 to
leads to a successive improvement of the array’s power absorption ability, as
global maxima become equal to
kW/m
2 (at
rad/s) and
kW/m
2 (at
rad/s) for
and
, respectively. Moreover, the frequency range, where a significant amount of power is absorbed, becomes more and more wider (i.e.,
rad/s
rad/s for
and
rad/s
rad/s for
). By further increasing
up to
, the power absorption ability of the array is successively reduced as compared with the case of
(
maxima are approximately equal to
kW/m
2 and
kW/m
2 for
and
, respectively) and it is realized at a slightly smaller frequency ranges (i.e., at
rad/s
rad/s for
and at
rad/s
rad/s for
). For
, adequate power is also absorbed at
rad/s, where additional
peaks are observed. However, by increasing
, the corresponding power absorption ability of the array is successively reduced, and it is bounded at less wide frequency ranges. Taking all the above into consideration, it can be concluded that for a wall with
, the placement of the examined five-body array with
at a non-dimensional distance from the wall,
, equal to 3.0 leads to the best power absorption. Finally, compared to the isolated array, it is clear that the existence of the wall boundary positively affects the power absorption ability of the array leading to a significant increase of
at specific frequency ranges, as well as to more than one
peak, depending upon the value of
.
The results of
Figure 9 demonstrate that
is a critical design parameter that directly affects the power absorption ability of the examined five-body array. For the smallest examined non-dimensional distance from the wall, the power absorption ability of the array is not driven by resonance phenomena, as significant heave exciting forces and responses, resulting from the hydrodynamic interactions of the WECs with the wall boundary, exist at wave frequencies outside the frequency range (
rad/s
rad/s), where resonance phenomena occur. However, the opposite holds true for the rest of the examined
values, where maximum
values occur at
rad/s
rad/s, since the presence of the wall boundary does not have a negative effect on the WECs’ heave exciting forces in this frequency range, and thus on the relevant
amplification due to resonance. Up to
the placement of the array at successively larger distances from the wall induces hydrodynamic interactions between the WECs and the boundary that enhance consecutively the hydrodynamic behavior of the WECs, and thus the array’s power absorption ability. However, a further increase of
to values larger than
does not enable to exploit in the best possible way the disturbances induced from the wall wave, and thus it leads to arrays with consecutively reduced power absorption ability. Consequently, among the examined
values,
is considered to be the upper limit of this design parameter, in terms of power absorption enhancement.
3.4. Effect of the Length of the Wall on the Array’s Performance
The effect of the length of the wall on the hydrodynamic behavior of the semi-immersed oblate spheroidal WECs is shown in
Figure 10 and
Figure 11, where the variations of
and
for all WECs in the array as a function of
are presented, respectively, for
, and
, as well as well as for the isolated array.
Starting with the heave exciting forces (
Figure 10),
for all WECs and for
varies in a similar manner as in the case of
and
, already described in
Section 3.2 and
Section 3.3, respectively. However, for this
value, a second peak at
rad/s occurs for WEC
j,
, and
(
Figure 10b,c). The increase of
leads to a more rapid increase of
, from the limiting value of
at
rad/s, and thus to a shift of the
first peaks (global maxima) at smaller wave frequencies. Moreover, it results to larger values of the aforementioned
peaks. The above trends introduce differences in the values of
at
rad/s
rad/s in the case of WEC
j,
, and
(
Figure 10a,c) and at
rad/s
rad/s in the case of WEC
j,
and
(
Figure 10b). Compared to the isolated array, it is straightforward that the presence of the wall, irrespectively of its length, affects the WECs’ heave exciting forces in the whole examined frequency range, since for all
values examined,
does not demonstrate a continuous smooth decrease as in the case of the isolated array. This in turn leads to different
values, especially at
rad/s
rad/s, where for
a great increase of the heave exciting forces occurs as compared with the isolated array.
With regards to WECs’ heave responses (
Figure 11),
for all WECs and for
varies in a similar manner as in the case of
and
already described in
Section 3.2 and
Section 3.3, respectively. Nevertheless,
for WEC
j,
, and
(
Figure 11b,c) depicts a second peak at
rad/s in absolute accordance with the corresponding heave exciting forces. On the one hand, for the outer WECs of the array (
Figure 11a), slightly larger
values compared to
for both
and
are observed at
rad/s
rad/s, where resonance phenomena occur, since the corresponding
values also become larger at this frequency range (
Figure 10a). On the other hand, for all WECs, the increase of
leads to larger
values at
rad/s
rad/s, i.e., in the low frequency range, in accordance to the results of
Figure 10. Finally, compared to the isolated array, the wall boundary, irrespectively of its length, significantly increases the WECs’ responses at
rad/s
rad/s and at
rad/s
rad/s, whereas the opposite holds true at
rad/s
rad/s.
The effect of
on the mean power absorbed by the whole array is shown in
Figure 12, where
is plotted as a function of
for
and
, as well as for the case of the isolated array. Since for all
cases examined, the array is placed at a distance
from the wall, the power absorption ability of the array is mainly driven by the WECs’ resonance. Therefore, maximum
values occur at
rad/s
rad/s, where resonance phenomena occur. However, the presence of the wall boundary introduces a second peak in the vicinity of
rad/s due to the existence of significant heave exciting forces, and thus large
values in the examined low frequency range, as previously described. Compared to
, where the maximum value of
is approximately equal to
kW/m
2, the increase of
to
enhances, to a small extent, the power absorption ability of the array, since the maximum value of
becomes equal to
kW/m
2. A further increase of
does not lead to any improvement of the array’s power absorption ability as compared with
. As for
rad/s
rad/s, where the second peak of
is observed, the change of
has an insignificant effect on the values of
. Finally, compared to the isolated array, it is clear that the existence of the wall boundary positively affects the power absorption ability of the array, as it results in a significant increase of
, especially in the frequency range, where resonance phenomena occur.
3.5. Spatial Variation of the Diffracted Wave Field
For physically interpreting the existence of great differences of the heave exciting forces at specific frequency ranges for different distances of the array from the wall (
Figure 7), the spatial variation of the non-dimensional wave elevation,
, due to only diffracted waves, i.e., first term in Equation (14), is taken into account. Specifically, for the array of the semi-immersed oblate spheroidal WECs with
placed in front of a wall of
,
Figure 13 and
Figure 14 show the
spatial variation of the diffracted wave field indicatively for
and
calculated at
and
for two characteristic values of
, equal to
rad/s and
rad/s, respectively. At
rad/s, the WECs’ heave exciting forces for
show significant values as compared with
, while the opposite holds for
rad/s, where, moreover,
for all WECs in the case of
obtains a characteristic local minimum (
Figure 7). For the above two wave frequencies, the diffracted wave field in the seaward side of the wall in the absence of the WECs is also shown in
Figure 15, which more clearly demonstrates the wave disturbances induced by the wall itself.
Starting with the case of
rad/s, the presence of the wall boundary in the absence of the WECs (
Figure 15a), at approximately
, leads to the formation of a longitudinal zone, with a sinusoidal-like variation pattern along
(coined hereafter “zone A1”), where
has values almost equal to zero. Outside of this zone,
varies quite intensively and has significant values. By placing the WECs in front of the wall (
Figure 13), zone A1 still exists; depending, however, upon the examined
value, the zone’s width along
and its variation pattern along
are modified due to the hydrodynamic interactions among the WECs and between the wall and the WECs. Accordingly, from a physical point of view, the case of
(
Figure 13a) corresponds to the placement of WECs outside of zone A1. More specifically, the WECs are placed in an area between the leeward side of zone A1 and the wall, where, the combined diffraction disturbances induced by the WECs and the wall amplify the intense variation pattern and the large values of
already existing, due to the presence of the wall. As a result, a non-symmetric (with respect to the WECs’ local horizontal axes) diffracted wave field, with significant
values, is formed around the WECs. On the other hand, the case of
(
Figure 13b) is physically equivalent to the placement of the WECs inside zone A1. This in turn, leads to the formation of an almost symmetric (with respect to the WECs’ local horizontal axes) diffracted wave field around the WECs, with quite small
values. All the above advocate the existence of larger
values for
at
rad/s as compared with the case of
, in absolute accordance with the results of
Figure 7.
In the case of
rad/s, the consideration of incident waves with shorter wave length results in the formation of three distinct longitudinal zones that are almost parallel to the wall, at the seaward side of the boundary (
Figure 15b), where
has almost zero values. The first zone (coined hereafter “zone B1”) is bounded at approximately
, while the second and the third zones (coined hereafter “zone B2” and “zone B3”, respectively) are bounded at approximately
and
, respectively. These zones remain almost unaffected, when the WECs are placed close to the wall, i.e., in the case of
(
Figure 14a), whereas for
the existence of the WECs modifies the width and the variation pattern of these zones along
(
Figure 14b). Following a rationale similar to the case of
Figure 13, from a physical point of view, the non-dimensional distance from the wall
(
Figure 14a) corresponds to the placement of WECs inside zone B1. This, in turn, results in the existence of an almost symmetric (with respect to the WECs’ local horizontal axes) diffracted wave field around the WECs, with quite small
values. However, the case of
(
Figure 14b) is physically equivalent with the placement of the WECs in the area between zones B1 and B2, where intense variation patterns and significant values of
are observed. Consequently, a non-symmetric (with respect to the WECs’ local horizontal axes) diffracted wave field with significant
values is formed around the WECs. All the above advocate for the existence of very small
values for
at
rad/s as compared with
, in accordance with the results of
Figure 7.