# Hydrodynamic Investigation of a Dual-Cylindrical OWC Wave Energy Converter Integrated into a Fixed Caisson Breakwater

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{1}and D

_{2}, respectively. A three-dimensional Cartesian coordinate system (o-xyz) is defined with the center of origin o locating at the cross-point of the undisturbed water plane, x-axis directing along the propagation of the incident waves, and z-axis orientating vertical upwards as the positive direction.

#### 2.1. Power Take-off Model

_{A}is the absorbed power of the OWC device, and E

_{I}is the flux of the incidence wave energy, which can be theoretically calculated [21,22]

_{A}can be obtained by the integration equation [21,22]

_{A}can be written as [23]

_{φ}is the sectional area of the air chamber, V

_{φ}(t) is the vertical speed of the water surface in the air chamber, B

_{a}is the damping coefficient.

#### 2.2. Boundary Value Problem

_{1}is defined by r ≥ R

_{2}and −d ≤ z ≤ 0, the internal subdomain Ω

_{2}is by R

_{1}≤ r ≤ R

_{2}and −d

_{1}≤ z ≤ 0, respectively. Correspondingly, the Φ

_{1}and Φ

_{2}represent the velocity potentials in Ω

_{1}and Ω

_{2}.

**Re**[ ] denotes taking the real part of a complex equation.

_{D}, S

_{F}, S

_{B}represent the seabed, external free surface and mean wet body surface, respectively.

_{1}and Φ

_{2}can be written as:

_{2}, the corresponding boundary value problem can be written as follows:

_{1}is the height from free surface to the lower edge of the inlet, s is the submergence height of the top edge of the inlet and $\partial /\partial r$ denotes the radial partial derivative of the variable.

#### 2.3. Mathematical Solutions

_{mn}, B

_{mn}and C

_{mn}are the constant coefficients, ε

_{m}is the Neumann symbol as ε

_{m}= {1, m = 0; 2, m ≥ 0}, $E=-igA/\omega $ with A as the wave amplitude, g the gravitational acceleration, ω the angular frequency, J

_{m}the Primal Bessel function, K

_{m}the modified Bessel function of the second kind, H

_{m}the Hankel function, H

_{m}

^{(1)}the Primal Hankel function, H

_{m}

^{(2)}the Hankel function of the second-kind and I

_{m}the Primal modified Bessel function, all of order m. Expressions of P

_{mn}and Q

_{mn}are shown as follows:

_{n}is the positive real root of Equation (27), the vertical Eigen-function Z

_{n}(z) can be obtained according to the boundary conditions and expressed as

_{1}and Ω

_{2}applied on the adjacent subdomains interfaces and body surface boundary conditions can be rewritten as:

_{1}and Φ

_{2}into the continuous matching conditions (Equations (29)–(33)), the following equations containing the coefficients of the unknown terms can be rewritten as:

_{0n}and a

_{nn}can be expressed as:

## 3. Results and Discussion

#### 3.1. Validation

_{2}/d =1.0 and 2.0 were considered to compare the hydrodynamic force between the analytical solution and the numerical results [24]. In the present analytical model, the height of the semi-arc inlet and the angle between the two baffles were set to zero, the wave height H is 0.06 m and the water depth d is 0.3 m. Figure 3 shows the comparison results of the horizontal wave force and wave moment with different kd. It can be seen that results calculated by the present analytical solution model have a good agreement with the results obtained by MacCamy and Fuchs [24]. The maximum differences of the wave force and the wave moment between the two methods are 1.4% and 3.3%, respectively.

_{1}/D

_{2}= 0.3 and the baffle wall angle θ = 200° was carried out, the results are shown in Figure 4. It can be found that the proposed dual-cylinder OWC device (with the normal case of D

_{1}/D

_{2}= 0.3 and θ = 200°) can obviously improve the efficiency of wave energy conversion. And for further study of the improvement of the power extraction efficiency, the impact of the OWC geometrical properties will be discussed in Section 3.

_{1}= 0.225 m, R

_{1}/R

_{2}= 0.5, θ = 180° are fixed in the validation. Two different wave heights of H = 0.03 m and H = 0.06 m are considered in the analytical model. The wave energy conversion efficiency of the device with different wave height ratios H

_{a}/H was calculated in the range of T = 0.791–1.107s, allowing the wave conditions to keep consistent with the experimental tests [20]. Comparisons between the analytical solution and the experimental results are presented in Figure 5. Black and red diamond markers indicate the results of the energy conversion efficiency (the left axis), and the blue and red circle markers represent the H

_{a}/H values (the right axis). The analytical model shows a good agreement with the measured data, with the mean relative error of ξ, and H

_{a}/H between the measured and calculated values are 3.68% and 4.8% for H = 0.06 m, and 8.45% and 7.92% for H = 0.03 m, respectively. The larger difference at small wave periods (see Figure 5) could be due to the ignorance of the air pressure force in the air nozzle area in the analytical model.

#### 3.2. Hydrodynamic Characteristics Inside the OWC Chamber

_{1}/h = 0.38, h = 0.5 m and H = 0.06 m. The incident wave angle is 0° between the x-axis and the wave propagation direction. Figure 6 shows the time series of the wave surface oscillating at four positions A (−D

_{2}/4 − D

_{1}/4, 0), B (−$\sqrt{2}$D

_{2}/8 − $\sqrt{2}$D

_{1}/8, $\sqrt{2}$D

_{2}/8 + $\sqrt{2}$D

_{1}/8), C (0, −D

_{2}/4 − D

_{1}/4) and D (−D

_{2}/2, 0) inside the chamber. Three non-dimensional wave number kd with different geometric characteristics for typical cases of C

_{1}(D

_{1}/D

_{2}= 0.7, kd = 0.8756), C

_{2}(D

_{1}/D

_{2}= 0.7, kd = 1.9641) and C

_{3}(D

_{1}/D

_{2}= 0.5, kd = 1.9641) are calculated to investigate the wave height variation in the chamber of the device.

_{2}and C

_{3}compared with condition C

_{1}due to the larger dimensionless wave number kd; the water motions at points A–D are mostly in phase in the air chamber. When the inside chamber volume is smaller (D

_{1}/D

_{2}= 0.7), the water elevation amplitude at point D is close to that at point A for shorter waves, as shown in Figure 7b.

_{a}inside the chamber is normalized by the incidence wave height H. The analytical results of the dimensionless parameter H

_{a}/H with different wave periods are shown in Figure 8. Four different incident wave heights, H/d = 0.1, 0.13, 0.17, and 0.2, are considered in the analytical model. It can be seen from the figure that the wave height of the free-surface inside the chamber increases with the increasing wave period until reaching a local maximum at T = 0.85 s, and then starts to decrease. This is because the conversion wave energy increases with the increment of the wave period for shorter waves, which results in a higher wave elevation inside the air chamber. However, when the water surface elevation increases to a certain extent, the air volume of the upper part in the chamber reduces evidently, resulting in stronger compression of the air and increased the pressure above the water surface, which leads to a restriction for the growing wave elevation inside the chamber. This illustrates that the cylindrical OWC device has the optimum conversion efficiency at a certain incident wave period. Meanwhile, for a fixed wave period condition, the relative wave height H

_{a}/H increases with the decrease of the incident wave height. Again, an explanation can be made that the larger wave height increases the pressure intensity inside the chamber, and then the compression of the air will perform a reaction formation on the water surface. It can also be observed in the experimental results obtained in our previous work [20].

#### 3.3. Effects of Relative Diameter on Conversion Efficiency and Hydrodynamic Loads

_{2}= 380 mm and s/h = 0.3, respectively. The incident wave height is H = 0.06 m in the analytical model. Four cylinder diameters D

_{1}/D

_{2}= 0.1, 0.2, 0.4, and 0.7 are considered as the typical conditions to calculate the conversion efficiency of the OWC device for the considered range of kd.

_{1}/D

_{2}, the optimal conversion efficiency of the OWC has its unique corresponding kd. As shown in the Figure, this particular wave excitation period is related to the geometry parameter D

_{1}/D

_{2}. Additionally, it also can be found that the corresponding kd for the maximal efficiency increases with the increase of the ratio D

_{1}/D

_{2}(the chamber volume decrease). In other words, with the increase of the ratio D

_{1}/D

_{2}, the peak value caused by the resonance mode shifts towards the shorter period region.

_{1}/D

_{2}= 0.4. The results of the calculated conversion efficiency against different wave number kd for D

_{1}/D

_{2}= 0.1 and 0.2 are similar, and the peak value of the conversion efficiency ξ at D

_{1}/D

_{2}= 0.2 is relatively close to that at D

_{1}/D

_{2}= 0.7. This is due to the fact that when the area of the air inlet S

_{0}at the top of the chamber is fixed, the larger air chamber volume between the two cylinders could be regarded as an approximately enclosed space; in this case, most of the absorbed wave energy is used for the air compression in the air chamber. When D

_{1}/D

_{2}is comparatively large (for a small chamber), the efficiency of the wave extraction is low and only 22% wave energy could be converted for power generation. Therefore, in order to possess a better power capacity of the proposed OWC, the theoretical optimal geometry parameter with D

_{1}/D

_{2}= 0.4 is recommended within this considered range of D

_{1}/D

_{2}= 0.1–0.7 in the analytical model.

_{x}and wave moment m

_{y}of the outer cylinder with four relative diameters D

_{1}/D

_{2}. In Figure 10, the results indicate that both the wave force and the wave moment have a similar trend with the increasing kd. The wave loads go up with the increase of the wave number in the low-frequency region and then show a decreasing trend. In the high-frequency zone, compared with the condition D

_{1}/D

_{2}= 0.1, the decrease of the wave loads for the case D

_{1}/D

_{2}= 0.2 and 0.7 drop faster with the increasing kd, but for D

_{1}/D

_{2}= 0.4, there occurs an increasing trend yet again in the range of kd = 4–6, which may be due to the large wave extraction caused by the resonant mode.

#### 3.4. Effect of the Angle between the Baffle Wall

_{1}/D

_{2}, the height of the inlet s/h and the incident wave height H are set constant. Figure 11 shows the conversion efficiency of the device against different kd with four baffle angles, i.e., θ = 240°, θ = 180°, θ = 120°, and θ = 90°. It can be observed that the conversion efficiency ξ increases with the increasing baffle angle for θ = 90–180° and reaches a maximum value at θ = 180°. This is because with the increase of the baffle wall angle, the frontal width of the chamber and the volume size are getting larger, which results in a large number of waves flow into the air chamber for power extraction. Compared with the condition of baffle angle θ = 240°, the corresponding kd for the peak value of the conversion efficiency with θ = 90–180° are relatively large and increases with the decreasing baffle angle θ. However, when the angle θ is greater than 180°, the maximal conversion efficiency ξ decreases, which may be the cause of the reflection of the waves acting on the baffle wall of the chamber. For the baffle angle θ less than 180°, the direction of wave reflections point into the inside of the chamber, which can drive up the wave surface in the air chamber, whilst for the baffle angle larger than 180°, the propagation of the reflected waves diffuse to the outside of the chamber so that the water heave motions in the chamber weakened and the conversion efficiency of the wave energy decreased.

_{x}and the bending moment m

_{y}increase with the increment of the baffle wall angle, and the hydrodynamic loads of the device have a similar trend with the increasing wave number kd. The largest difference of the wave force and the bending moment with four baffle angle θ = 240°, θ = 180°, θ = 120°, and θ = 90° occur at kd = 1.77 and kd = 1.61, respectively. In the high-frequency zone, the hydrodynamic load difference among the considered four cases decreased with the increasing kd. Finally, the values of wave force f

_{x}and the bending moment m

_{y}with different baffle angles are getting close to each other for larger wave number kd. In Figure 13, both the wave force and the wave moment have an increasing trend with the increase of kd. The results of the force and the moment against different wave number kd for θ = 240° and 120° are similar. And the force and moment for θ = 180° are the largest, which may be due to the resonant mode in the air chamber. On the other hand, for the inner cylinder, there is a different trend of the wave force and the bending moment acting on the outer cylinder. This may be the cause of the seriously oscillatory heave motion of the waves in the inner air chamber, which increases the power extraction but decrease the horizontal wave force and the bending moment on the inner cylinder.

#### 3.5. Comparison of the Geometry Parameters

_{1}/D

_{2}= 0.2–0.75. The chamber inlet height and the incident wave height are set as constant s/h = 0.3 and H/d = 0.2, respectively. Figure 14 presents the energy conversion efficiency of the OWC device with six typical cases, i.e., T

_{1}(θ = 180°, D

_{1}/D

_{2}= 0.4), T

_{2}(θ = 160°, D

_{1}/D

_{2}= 0.45), T

_{3}(θ = 200°, D

_{1}/D

_{2}= 0.35), T

_{4}(θ = 240°, D

_{1}/D

_{2}= 0.2), T

_{5}(θ = 90°, D

_{1}/D

_{2}= 0.7) and T

_{6}(θ = 90°, D

_{1}/D

_{2}= 0.75). It can be seen that the maximum conversion efficiency occurs at the geometric parameter condition of D

_{1}/D

_{2}= 0.4 and θ = 180° (Case T

_{1}), and conversely, the lowest conversion efficiency is Case T

_{6}, and for Cases T

_{2}-T

_{5}, the wave energy conversion efficiency of the OWC decreased in order. This illustrates that compared with the factor of baffle angle θ, the relative diameter ratio of the dual cylinders plays a dominant role in improving the power extraction capacity of the OWC. It also can be observed that the corresponding kd with the optimal conversion efficiency ξ for Case T

_{6}is the largest, followed by the Cases of T

_{5}, T

_{2}, T

_{1}, T

_{3}and T

_{4}, which rank with the same increasing sequence of the baffle angle. It demonstrates that the proposed OWC device has better absorbability for longer waves with larger baffle angle, and better absorption and conversion efficiency for shorter waves under smaller baffle angle conditions. In other words, the angle of the baffle wall in the chamber has a significant role to play in possessing better efficiency for the specific wave excitation periods. Therefore, the condition of Case T

_{1}is recommended as the optimal geometry parameters for this proposed OWC device to possess a better capacity of the wave power extraction.

## 4. Conclusions

- The water surface elevation inside the chamber increases with the increasing wave period until it reaches a local maximum at a certain period (i.e., T = 0.85 s in the current study) and then starts to decrease.
- The conversion efficiency of the OWC device for different relative diameters and the baffle wall angles increased with the increasing wave number kd in the low-frequency zone. The corresponding kd for the optimal conversion efficiency of the OWC shifts towards the shorter period region with the increase of the relative diameter D
_{1}/D_{2}. - Given the same wave and geometry condition, the optimal conversion efficiency occurs when the relative cylindrical diameter D
_{1}/D_{2}is 0.4 and the baffle wall angle is 180°. It is hence concluded that the theoretical optimal geometry parameters as D_{1}/D_{2}= 0.4 and θ = 180° are recommended for a better capacity of wave power extraction. - The wave loads of the whole OWC go up with the increase of the wave number and then shows a fast decreasing trend in high-frequency regions.
- Compared with baffle-wall angle θ, the diameter ratio D
_{1}/D_{2}of the dual cylinders plays a dominant role in increasing the wave energy conversion efficiency. While for a specific incident wave period, the power extraction capacity of the OWC mainly determined by the angle of the baffle wall in the chamber.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparisons of the wave force and wave moment between the present model and MacCamy’s results.

**Figure 4.**Comparison of the wave energy conversion efficiency ξ between dual-cylinder type OWC and hollow-cylindrical OWC.

**Figure 5.**Comparison of the conversion efficiency ξ with different H

_{a}/H between the present model and experimental data.

**Figure 7.**Time series of wave free-surface elevation at four positions in the chamber at (

**a**) C

_{1}, (

**b**) C

_{2}, and (

**c**) C

_{3}conditions.

**Figure 9.**Effects of cylinder diameter D

_{1}/D

_{2}on power extraction efficiency with different kd.

**Figure 10.**Distribution of wave force and wave moment on the outer cylinder with different D

_{1}/D

_{2}.

**Figure 12.**Variation of the wave force and the bending moment on the outer cylinder with different baffle wall angles.

**Figure 13.**Variation of the wave force and the bending moment on the inner cylinder with different baffle wall angles.

**Figure 14.**Variation and range extended comparison of the OWC conversion efficiency for typical cases.

Dimensional Variable | Physical Unit |
---|---|

Wave amplitude, A | m |

Water depth, d | m |

Diameter of the outer cylinder, D_{1} | mm |

Diameter of the inner cylinder, D_{2} | mm |

Gravitational acceleration, g | m∙s^{−2} |

Height of the cylinder, h | mm |

Incident wave height, H | m |

Incident wave number, k | -- |

Air pressure, P_{0} | kPa |

Height of the opening, s | mm |

Time, t | s |

Velocity potential, Φ | m∙s^{−1} |

Wave period, T | s |

Angular frequency, ω | rad∙s^{−1} |

Water density, ρ | kg∙m^{−3} |

The angle between partition walls, θ | -- |

Power extraction efficiency, ξ | % |

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## Share and Cite

**MDPI and ACS Style**

Wan, C.; Yang, C.; Fang, Q.; You, Z.; Geng, J.; Wang, Y.
Hydrodynamic Investigation of a Dual-Cylindrical OWC Wave Energy Converter Integrated into a Fixed Caisson Breakwater. *Energies* **2020**, *13*, 896.
https://doi.org/10.3390/en13040896

**AMA Style**

Wan C, Yang C, Fang Q, You Z, Geng J, Wang Y.
Hydrodynamic Investigation of a Dual-Cylindrical OWC Wave Energy Converter Integrated into a Fixed Caisson Breakwater. *Energies*. 2020; 13(4):896.
https://doi.org/10.3390/en13040896

**Chicago/Turabian Style**

Wan, Chang, Can Yang, Qinghe Fang, Zaijin You, Jing Geng, and Yongxue Wang.
2020. "Hydrodynamic Investigation of a Dual-Cylindrical OWC Wave Energy Converter Integrated into a Fixed Caisson Breakwater" *Energies* 13, no. 4: 896.
https://doi.org/10.3390/en13040896