# Performance of an Array of Oblate Spheroidal Heaving Wave Energy Converters in Front of a Wall

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Modeling

## 3. Results and Discussion

#### 3.1. General Remarks

#### 3.2. Comparison of Arrays with Different WEC Geometries

#### 3.3. Effect of WECs’ Distance from the Wall on the Array’s Performance

^{2}. Moreover, additional $p\left(\omega \right)$ peaks with smaller values occur at $\omega \text{}=\text{}2.5$ rad/s and $\omega \text{}=\text{}3.1$ rad/s as a result of the relevant $RA{O}_{3}$ peak values (Figure 8). By increasing $c/a\text{}$to $2.0$, a significant amount of power is absorbed at larger wave frequencies and, more specifically, at $2.1$ rad/s $<\text{}\omega \text{}2.7$ rad/s, where resonance phenomena also occur. Moreover, the power absorption ability of the array is substantially enhanced, since the maximum value of $p\left(\omega \right)$, occurring at $\omega \text{}=\text{}2.45$ rad/s, is approximately equal to $270$ kW/m

^{2}. Similar behavior is observed for the rest of the examined $c/a$ values. However, the increase of $c/a\text{}$from 2.0 to $3.0$ leads to a successive improvement of the array’s power absorption ability, as $p\left(\omega \right)$ global maxima become equal to $370$ kW/m

^{2}(at $\omega =2.4$ rad/s) and $385$ kW/m

^{2}(at $\omega \text{}=\text{}2.3$ rad/s) for $\text{}c/a=\text{}2.5$ and $c/a\text{}=\text{}3.0$, respectively. Moreover, the frequency range, where a significant amount of power is absorbed, becomes more and more wider (i.e., $1.9$ rad/s $<\text{}\omega \text{}\text{}2.7$ rad/s for $c/a=\text{}2.5$ and $1.7$ rad/s $<\text{}\omega \text{}2.7$ rad/s for $c/a=\text{}3.0$). By further increasing $c/a$ up to $4.0$, the power absorption ability of the array is successively reduced as compared with the case of $c/a=\text{}3.0$ ($p\left(\omega \right)$ maxima are approximately equal to $380$ kW/m

^{2}and $340$ kW/m

^{2}for $\text{}c/a=\text{}3.5$ and $\text{}c/a=\text{}4.0$, respectively) and it is realized at a slightly smaller frequency ranges (i.e., at $1.6$ rad/s $<\text{}\omega \text{}2.5$ rad/s for $\text{}c/a=\text{}3.5$ and at $1.5$ rad/s $<\text{}\omega \text{}\text{}2.4$ rad/s for $\text{}c/a=\text{}4.0$). For $\text{}c/a\ge \text{}2.0$, adequate power is also absorbed at $\omega <1.5$ rad/s, where additional $p\left(\omega \right)$ peaks are observed. However, by increasing $c/a$, 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 ${l}_{w}/a\text{}=\text{}36$, the placement of the examined five-body array with ${l}_{bet}/a=\text{}4$ at a non-dimensional distance from the wall, $c/a$, 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 $p\left(\omega \right)$ at specific frequency ranges, as well as to more than one $p\left(\omega \right)$ peak, depending upon the value of $c/a$.

#### 3.4. Effect of the Length of the Wall on the Array’s Performance

^{2}, the increase of ${l}_{w}/a$ to $36$ enhances, to a small extent, the power absorption ability of the array, since the maximum value of $p\left(\omega \right)$ becomes equal to $385$ kW/m

^{2}. A further increase of ${l}_{w}/a$ does not lead to any improvement of the array’s power absorption ability as compared with ${l}_{w}/a\text{}=\text{}36$. As for $0.5$ rad/s $<\text{}\omega \text{}\text{}1.3$ rad/s, where the second peak of $p\left(\omega \right)$ is observed, the change of ${l}_{w}/a$ has an insignificant effect on the values of $p\left(\omega \right)$. 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 $p\left(\omega \right)$, especially in the frequency range, where resonance phenomena occur.

#### 3.5. Spatial Variation of the Diffracted Wave Field

## 4. Conclusions

- Compared to the cylindrical WECs, the deployment of either the hemisphere-shaped or the oblate spheroidal WECs improves the power absorption ability of the array to a great extent. However, among all the three WECs’ geometries examined, the oblate spheroidal WECs are considered to have the best power absorption ability due to their intrinsic hydrodynamic characteristics that also enable the consideration of larger ${b}_{PTO}$ values.
- Irrespectively of the values of ${l}_{w}/a$ and/or $c/a$, the presence of the wall boundary in the leeward side of the oblate spheroidal WECs, contrary to the isolated array, leads to the existence of ${F}_{3}$ global maxima at $\omega \ne 0.01$ rad/s with values larger than 1.0.
- For different $c/a$ values, the oblate spheroids’ heave exciting forces show great differences at specific frequency ranges. From a physical point of view, this fact can be related to the realization or not of an array’s placement at existing characteristic wave field zones, resulting from the presence of the wall, where the relevant wave elevation has values almost equal to zero. In cases, where $c/a$ falls within these zones, a smooth diffracted wave field around the WECs with symmetrical features is formed advocating the existence of very small ${F}_{3}$ values, while the opposite holds true when the WECs are placed outside of these zones.
- For the smallest examined distance from the wall, the power absorption ability of the oblate spheroidal WECs array is not driven by resonance phenomena, as large heave exciting forces, significant responses, and thus maximum $p\left(\omega \right)$ values, occur at wave frequencies outside the range, where WECs’ resonance occurs. The opposite holds true for the remaining examined$\text{}c/a$ values, where the presence of the wall does not impose any restrictions on the $RA{O}_{3}$ amplification due to resonance.
- The placement of the oblate spheroidal WECs array at successively larger distances from the wall induces hydrodynamic interactions between the spheroids and the boundary that enhance consecutively the hydrodynamic behavior of the WECs, and thus the array’s power absorption ability. However, this holds true for $c/a$ up to $3.0,$ since a further increase of $c/a$ leads to arrays with consecutively reduced power absorption ability. Consequently, among the $c/a$ values examined in this paper, $c/a=3.0$ presents an upper limit of this design parameter in terms of power absorption enhancement. Compared to the isolated array, the presence of the wall boundary positively affects the power absorption ability of the array leading to a significant increase of $p\left(\omega \right)$ at specific frequency ranges, as well as to more than one $p\left(\omega \right)$ peak, depending upon the value of $c/a$.
- The increase of ${l}_{w}/a$ from 18 to $36$ enhances the power absorption ability of the oblate spheroidal WECs array to a small extent, and a further increase of ${l}_{w}/a$ does not lead to any relevant improvement. Compared to the isolated array, the existence of the wall boundary, irrespectively of its length, affects positively the power absorption ability of the array, since it results in a significant increase of $p\left(\omega \right)$, especially at the frequency range where resonance phenomena occur.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Geometry of the investigated arrangement and definition of basic quantities: (

**a**) X-Y plane and (

**b**) X-Y plane.

**Figure 2.**Schematic representation of the wetted surfaces of the WECs’ geometries: (

**a**) Oblate spheroid, (

**b**) cylinder, and (

**c**) hemisphere.

**Figure 3.**Variation of non-dimensional hydrodynamic coefficients in heave for the WEC geometries considered (results correspond to single isolated bodies): (

**a**) Added mass and (

**b**) radiation damping.

**Figure 4.**Comparison of ${F}_{3}$ for arrays with different WEC geometries and for ${l}_{w}/a\text{}=\text{}18$, $c\text{}=\text{}6$ m, ${l}_{bet}/a=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 5.**Comparison of $RA{O}_{3}$ for arrays with different WEC geometries and for ${l}_{w}/a\text{}=\text{}18$, $c\text{}=\text{}6$ m, ${l}_{bet}/a\text{}=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 6.**Comparison of $p\left(\omega \right)$ for arrays with different WEC geometries: (

**a**) ${l}_{w}/a\text{}=\text{}18$ and (

**b**) ${l}_{w}/a\text{}=\text{}72$ ($c\text{}=\text{}6$ m, ${l}_{bet}/a\text{}=\text{}4$ for both ${l}_{w}/a$ cases).

**Figure 7.**Effect of $c/a$ on ${F}_{3}$ applied on the semi-immersed oblate spheroidal WECs for ${l}_{w}/a\text{}=\text{}36$ and ${l}_{bet}/a\text{}=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 8.**Effect of $c/a$ on $RA{O}_{3}$ of the semi-immersed oblate spheroidal WECs for ${l}_{w}/a\text{}=\text{}36$ and ${l}_{bet}/a\text{}=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 9.**Effect of $c/a$ on $p\left(\omega \right)$ of the array with semi-immersed oblate spheroidal WECs for ${l}_{w}/a\text{}=\text{}36$ and ${l}_{bet}/a\text{}=\text{}4$.

**Figure 10.**Effect of ${l}_{w}/a$ on ${F}_{3}$ applied on the semi-immersed oblate spheroidal WECs for $c/a=\text{}3$ and ${l}_{bet}/a\text{}=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 11.**Effect of ${l}_{w}/a$ on $RA{O}_{3}$ applied on the semi-immersed oblate spheroidal WECs for $c/a\text{}=\text{}3$ and ${l}_{bet}/a=\text{}4$: (

**a**) WEC1 and WEC5, (

**b**) WEC2 and WEC4, and (

**c**) WEC3.

**Figure 12.**Effect of ${l}_{w}/a$ on $p\left(\omega \right)$ of the array with semi-immersed oblate spheroidal WECs for $c/a=\text{}3$ and ${l}_{bet}/a=\text{}4$.

**Figure 13.**Spatial variation of ${\eta}_{D}/A$ at $\omega =1.5$ rad/s for the array with semi-immersed oblate spheroidal WECs (${l}_{w}/a=\text{}36$ and $\text{}{l}_{bet}/a=\text{}4$): (

**a**) $\text{}c/a=\text{}1.5$ and (

**b**) $\text{}c/a=\text{}3.0$.

**Figure 14.**Spatial variation of ${\eta}_{D}/A$ at $\omega \text{}=\text{}2.3$ rad/s for the array with semi-immersed oblate spheroidal WECs (${l}_{w}/a=\text{}36$ and $\text{}{l}_{bet}/a=\text{}4$): (

**a**) $\text{}c/a=\text{}1.5$ and (

**b**) $\text{}c/a=\text{}3.0$.

**Figure 15.**Spatial variation of ${\eta}_{D}/A$ at the seaward side of the wall with ${l}_{w}/a=\text{}36$ in the absence of WECs: (

**a**) $\omega \text{}=\text{}1.5$ rad/s and (

**b**) $\omega \text{}=\text{}2.3$ rad/s.

Geometry/Shape | $\mathit{a}$ (m) | $\mathit{b}$ (m) | ${\mathit{b}}_{\mathit{P}\mathit{T}\mathit{O}}\text{}(\mathbf{Ns}/\mathbf{m})$ |
---|---|---|---|

Oblate spheroid | 2.0 | 1.7 | $\mathrm{10,322.20}$ |

Cylinder | 1.4 | 1.0 | $2572.46$ |

Hemisphere | 1.8 | 1.8 | $7111.86$ |

Examined Design Parameter | $\mathit{c}/\mathit{a}$ | ${\mathit{l}}_{\mathit{w}}/\mathit{a}$ | ${\mathit{l}}_{\mathit{e}\mathit{d}\mathit{g}\mathit{e}}/\mathit{a}$ | ${\mathit{l}}_{\mathit{b}\mathit{e}\mathit{t}}/\mathit{a}$ |
---|---|---|---|---|

Non-dimensional distance from the wall ($c/a)$ | $1.5$, $2$, $2.5$, $3$, $3.5$, $4$ | $36$ | $10$ | $4$ |

Non-dimensional wall length (${l}_{w}/a$) | $3$ | $18$, $36$, $72$ | $1$, $10$, $28$ | $4$ |

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**MDPI and ACS Style**

Loukogeorgaki, E.; Boufidi, I.; Chatjigeorgiou, I.K.
Performance of an Array of Oblate Spheroidal Heaving Wave Energy Converters in Front of a Wall. *Water* **2020**, *12*, 188.
https://doi.org/10.3390/w12010188

**AMA Style**

Loukogeorgaki E, Boufidi I, Chatjigeorgiou IK.
Performance of an Array of Oblate Spheroidal Heaving Wave Energy Converters in Front of a Wall. *Water*. 2020; 12(1):188.
https://doi.org/10.3390/w12010188

**Chicago/Turabian Style**

Loukogeorgaki, Eva, Ifigeneia Boufidi, and Ioannis K. Chatjigeorgiou.
2020. "Performance of an Array of Oblate Spheroidal Heaving Wave Energy Converters in Front of a Wall" *Water* 12, no. 1: 188.
https://doi.org/10.3390/w12010188