# Performance of an Array of Oscillating Water Column Devices in Front of a Fixed Vertical Breakwater

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Hydrodynamic Problem

_{c}. The radius of the interior, bottom seated, coaxial cylindrical body is denoted by c. In addition, the distance between the center of the closest to the wall converter and the breakwater is denoted by L

_{w}, whereas the distance between adjacent OWCs is by L

_{b}. Small amplitude harmonic waves (with angular frequency ω, wave height H, and wave length λ) are incident to the breakwater at an angle β. A global, right-handed Cartesian co-ordinate system O-xyz is introduced with origin located at the bottom of the breakwater, with its vertical axis Oz directed upwards, while N local cylindrical co-ordinate systems $\left({r}_{q},{\theta}_{q},z\right)$, q = 1, 2, …, N are defined with origins at the intersection $\left({X}_{q},{Y}_{q}\right)$ of the sea bottom with the vertical axis of symmetry of each converter. The geometric layout of the breakwater-OWC system is illustrated in Figure 1.

- (a)
- on the water free surface:$$\{\begin{array}{l}{\omega}^{2}{\phi}_{j}^{q}-g\frac{\partial {\phi}_{j}^{q}}{\partial z}=0,j=0,s;z=h;{r}_{q}\ge a;c\le {r}_{q}\le b\hfill \\ {\omega}^{2}{\phi}_{P}^{qp}-g\frac{\partial {\phi}_{P}^{qp}}{\partial z}=\{\begin{array}{c}0,{r}_{q}\ge a\\ -{\delta}_{q,p}\frac{i\omega}{\rho},c\le {r}_{q}\le b\end{array};\hfill \end{array}$$
- (b)
- on the sea bottom:$$\frac{\partial {\phi}_{k}^{i}}{\partial z}=0,i=p,qp;k=0,s,P;z=0;$$
- (c)
- kinematic condition on the mean device’s wetted surface:$$\{\begin{array}{l}\frac{\partial {\phi}_{s}^{q}}{\partial {n}^{q}}=-\frac{\partial {\phi}_{0}^{q}}{\partial {n}^{q}}\hfill \\ \frac{\partial {\phi}_{p}^{qp}}{\partial {n}^{q}}=0\hfill \end{array}$$

## 3. Array’s Efficiency

## 4. Numerical Results

#### 4.1. Test Cases

_{c}and the water depth equals to h = 7.14α. Concerning the inner cylindrical body, it is assumed to be sea bottom seated with a radius of c = 0.4α The OWCs’ air turbine characteristics are considered to be equal to the Λ

_{opt}value of a similar OWC in isolation condition at its pumping resonance wave frequency [51], whereas the distance between the center of the closest to the wall converter and the breakwater is L

_{w}, and the distance between adjacent OWCs is L

_{b}= 4α (see Figure 1 and Figure 2).

#### 4.2. Effect of the OWCs Orientation to the Breakwater

_{w}= 3α, whereas the distance between the bottom of the external torus and the seabed equals to h

_{c}= 6.14α.

^{5}/(kN.s)].

#### 4.3. Effect of the OWCs’ Distance from the Breakwater

_{c}= 6.14α. The examined distances from the wall equal to L

_{w}= 3α, 6α, 9α, 12α, and the wave heading angles to β = 0, π/6, π/4. Additionally, the air turbine coefficient inside the OWCs equals to 10.60 [m

^{5}/(kN.s)].

_{w}= 6α, 9α, 12α) oscillate around those of the L

_{w}= 3α case. It can be also seen that the larger the distance of the converters from the wall is, the stronger the oscillatory behavior of the air pressure RAOs. As far as the examined wave heading angles are concerned, it can be seen from the left side of Figure 7, that the inner air pressure graphs follow a similar pattern, regardless of the wave heading angle. Nevertheless, it should be mentioned that the wave numbers in which oscillations in the air pressure occur, are shifted to higher values as the wave heading angle increases. Concerning the efficiency of the array at several distances from the breakwater, it can be seen (right side of Figure 7) that the examined distances between the devices and the vertical wall cause the array’s absorbed power to increase at some kα values and to decrease at another range of wave number compared to the efficiency of five OWCs in isolation condition. This is due to the reflected waves from the wall, which cause a dominant oscillating behavior on the array’s absorbed power values. Moreover, as the distance of the OWC from the vertical wall increases, the earlier the oscillatory behavior of the wave absorbed power occurs. As far as the incident wave angles are concerned, it can be obtained that the wave numbers in which oscillations in the power efficiency occur are shifted to higher values as β increases. Finally, it can concluded that the efficiency results for the close to wall distances (i.e., L

_{w}= 3α, 6α) follow a more steady pattern, at every examined wave number, compared to the results of the large-distance cases (i.e., L

_{w}= 9α, 12α), of which they attain a large oscillatory behavior.

#### 4.4. Effect of the OWCs’ Draught

_{w}= 3α. The examined OWCs’ draughts equal to h

_{c}= 6.14α, 5.64α, 5.14α, 4.64α, and the wave heading angles to β = 0, π/6, π/4.

^{5}/(kN.s)] for each examined draught h

_{c}= 6.14α, 5.64α, 5.14α, 4.64α, respectively, for α = 1 m.

^{2}, (right column) for various examined OWC draughts and wave heading angles. From the depicted results, it can be seen that the draught of the OWCs affects the inner air pressure head as well as the array’s efficiency. Specifically, as the draught of the converter increases, the pumping resonance wave frequency is shifted at lower values, thus the corresponding wave numbers in which the inner air pressure values resonance are transferred to lower values too. Furthermore, it can be noted that as the draught of the OWC increases, the air pressure head inside the 1st OWC decreases. As far as the examined wave heading angles is concerned, it is depicted that β has a minor effect on the inner air pressure head for kα < 0.5. On the other hand, for kα > 0.5, the inner air pressure head inside the 1st OWC decreases as the wave heading angle increases.

#### 4.5. Effect of the Distance between the OWCs

_{w}= 3α, whereas the distance between the bottom of the external torus and the seabed equals to h

_{c}= 6.14α. The air turbine coefficient inside the OWCs equals to 10.60 [m

^{5}/(kN.s)]. Several distances between the converters of the array are examined, i.e., L

_{b}= 4α; 8α; 16α; 32α.

_{b}= 32α case are similar to the corresponding values of an isolated OWC in front of a breakwater (see Figure 4a). On the other hand, it can be seen that the q

_{f}factor is significantly affected by the distance between the devices, especially at higher wave frequencies, in which the q

_{f}values appear as an oscillatory behavior (i.e., kα > 0.7). This behavior becomes more pronounced as the distance between the adjacent OWCs increases. Contrary, for kα < 0.7, the efficiency of the array seems not to be affected by the distance between the devices.

## 5. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

N | Number of OWC devices |

h | Water depth |

α | OWC outer radius |

b | OWC inner radius |

h_{c} | Distance between the bottom of the external torus and the seabed |

c | Radius of the internal cylindrical body |

L_{w} | Distance between the closest to the wall OWC and the vertical wall |

L_{b} | Distance between adjacent OWCs |

β | Wave heading angle |

ω | Wave frequency |

H | Wave height |

λ | Wave length |

r_{k,}θ_{k},z_{k} | Local co-ordinate system of the k OWC |

Φ | Time harmonic complex velocity potential |

${\phi}_{0}$ | Velocity potential of the undisturbed incident harmonic wave |

${\phi}_{s}^{q}$ | Scattered velocity potential of the q OWC |

${\phi}_{D}^{q}$ | Diffraction velocity potential of the q OWC |

${\phi}_{P}^{qp}$ | Radiation velocity potential resulting from the inner air pressure in p OWC |

${p}_{in0}^{p}$ | Amplitude of the oscillating pressure head in the chamber of the p OWC |

g | Gravitational acceleration |

ρ | Water density |

${n}^{q}$ | Unit normal vector |

δ_{q,p} | Kronecker’s symbol |

${S}_{0}^{q}$ | Mean wetted surface of the q OWC |

I | The infinite ring element around the q OWC |

II | The ring element below the q OWC |

III | The ring element inside the chamber of the q OWC |

$Q\left(t\right)$ | Time dependent air volume flow |

${u}_{z}$ | Vertical velocity of the water surface in the OWC |

${S}_{i}$ | Cross-sectional area of the inner water surface inside the OWC |

${\mathrm{q}}_{D}^{q}$ | Diffraction volume flow of the q OWC |

${\mathrm{q}}_{p}^{qp}$ | Pressure-dependent volume flow of the q OWC |

${g}_{p}^{qp}$ | Radiation conductance of the q OWC |

${f}_{p}^{qp}$ | Radiation susceptance of the q OWC |

Λ | Complex pneumatic admittance (air turbine coefficient) |

${\mathsf{\Lambda}}_{opt}$ | Air turbine coefficient optimum value |

$P\left(\omega \right)$ | Absorbed wave power by each OWC of the array |

${q}_{f}$ | q-factor term |

## Appendix A

_{m}and K

_{m}are the m-th order Hankel function of the first kind and the modified Bessel function of the second type, respectively, while J

_{m}is the m-th order Bessel function of first kind. Additionally, ${\delta}_{D}=d;{\delta}_{P}=1$. The term ${Z}_{n}\left(z\right)$ denotes orthonormal function in [0, h] defined as follows:

_{m}is the m-th order modified Bessel function of first kind.

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**Figure 1.**3D representations of an OWC array in front of a breakwater: (

**a**) array of N-OWCs in front of a breakwater; (

**b**) schematic representation of the examined OWC, side view-xz.

**Figure 2.**2D plane representation of a breakwater-OWC system using the method of images. The image devices are denoted dashed.

**Figure 3.**Schematic representations of the considered OWC array arrangements in front of a vertical breakwater and the numbering of the devices’ considered: (

**a**) five examined configurations of OWCs arranged in parallel direction to the wall; (

**b**) five examined configurations of OWCs arranged perpendicularly to the wall; (

**c**) five examined configurations of OWCs arranged rectangularly to the wall.

**Figure 4.**Air pressure inside the 1st OWC (left column figures) and array’s ${q}_{f}$ factor (right column figures) versus kα for various examined wave heading angles: (

**a**) β = 0; (

**b**) β = π/6; (

**c**) β = π/4. Parallel arrangement.

**Figure 5.**Air pressure inside the 1st OWC (left column figures) and array’s ${q}_{f}$ factor (right column figures) versus kα for various examined wave heading angles: (

**a**) β = 0; (

**b**) β = π/6; (

**c**) β = π/4. Perpendicular arrangement.

**Figure 6.**Air pressure inside the 1st OWC (left column figures) and array’s ${q}_{f}$ factor (right column figures) versus kα for various examined wave heading angles: (

**a**) β = 0; (

**b**) β = π/6; (

**c**) β = π/4. Rectangular arrangement.

**Figure 7.**Air pressure inside the 1st OWC (left column figures) and array’s ${q}_{f}$ factor (right column figures) versus kα for various examined distances between the array and the breakwater and wave heading angles: (

**a**) β = 0; (

**b**) β = π/6; (

**c**) β = π/4.

**Figure 8.**Air pressure inside the 1st OWC (left column figures) and array’s absorbed power P(ω) (right column figures) versus kα for various examined OWC draughts and wave heading angles: (

**a**) β = 0; (

**b**) β = π/6; (

**c**) β = π/4.

**Figure 9.**Air pressure inside the 1st OWC (left column figure) and array’s ${q}_{f}$ factor (right column figure) versus kα for various examined distances between the bodies. Here, β = 0.

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**MDPI and ACS Style**

Konispoliatis, D.N.
Performance of an Array of Oscillating Water Column Devices in Front of a Fixed Vertical Breakwater. *J. Mar. Sci. Eng.* **2020**, *8*, 912.
https://doi.org/10.3390/jmse8110912

**AMA Style**

Konispoliatis DN.
Performance of an Array of Oscillating Water Column Devices in Front of a Fixed Vertical Breakwater. *Journal of Marine Science and Engineering*. 2020; 8(11):912.
https://doi.org/10.3390/jmse8110912

**Chicago/Turabian Style**

Konispoliatis, Dimitrios N.
2020. "Performance of an Array of Oscillating Water Column Devices in Front of a Fixed Vertical Breakwater" *Journal of Marine Science and Engineering* 8, no. 11: 912.
https://doi.org/10.3390/jmse8110912