# Hydrodynamic Performance of a Floating Offshore Oscillating Water Column Wave Energy Converter

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Computational Fluid Dynamic and Aerodynamic Wave Model

_{s}and X

_{s}are the length and distance from the starting point of the damping layers, respectively. The Petrov–Galerkin Finite Element Method (PG-FEM) code, originally developed by Zhao, et al. [54] and later extended for modelling waves, was utilised in this investigation to solve the RANS equations [47,49].

_{a}(t) is the air pressure in the OWC chamber, p

_{a0}is the atmospheric pressure, and K

_{t}is the turbine coefficient. The instantaneous power generated by the turbine P

_{T}is:

_{w}is the energy of the incoming waves calculated using the second order Stokes wave theory [59]:

_{i}is the incident wave height, ω is the angular frequency of the waves, k is the wave number, and h is the water depth.

#### 2.2. Wave-Induced Heave and Horizontal Motion

_{n}is the natural frequency measured in a vacuum, $\zeta $ is the damping ratio, and ${F}_{\mathrm{x}},$F

_{y}are the fluid force in the horizontal and vertical direction. The natural frequency is related to the stiffness of the mounting spring as ${f}_{\mathrm{n}}=\frac{1}{2\pi}\sqrt{K/m}$, where K and m are the stiffness of the mounting spring and the mass of the OWC structure, respectively. The vertical fluid force includes two components: the hydraulic force on the submerged walls of the OWC and the force on the ceiling of the OWC chamber caused by the air pressure.

## 3. Numerical Results

_{i}) = 0.04 m, the thickness of the OWC walls = 0.01 m, the draught of the front and rear walls is d = 0.1 m, i.e., d/h = 0.25, and the chamber length B = 0.18 m for both the vertical and horizontal motion of the OWC device. For a two-dimensional simulation, the width of the chambers is unit width, i.e., W = 1 m. The turbine coefficient is K

_{t}= 3000 Pa·m

^{−3}·s except in Figure 2, where a series of turbine coefficients are studied. The computational mesh used in this study has the same density as the one used by Mia et al. [52], who conducted a systematic mesh dependency study to prove that the mesh was sufficiently dense for converged results. Figure 1c shows the computational mesh near the OWC. The mesh is refined near the water surface and the surface of the OWC walls. The mesh dependency study will not be repeated here.

_{t}, the chamber width to wavelength ratio (B/L) has a significant impact on the performance of the OWC, where L is the wavelength. To determine the best K

_{t}and B/L for the highest power, a stationary OWC device, without any motion, is first simulated, with a draught height d = 0.1 m, B/L in the range of $0.095\le \frac{B}{L}\le 0.318$, and 15 turbine coefficients. Figure 2 shows that, for all turbine coefficients, there is a consistent pattern in the variation of the efficiency with B/L. For all K

_{t}values, the efficiency increases with an increase in B/L until they reach its maximum value. Further increase in the B/L results in a decrease in the efficiency. The maximum efficiency over the whole range of the B/L is defined as the best efficiency (${\epsilon}_{\mathrm{best}}$) for a particular turbine coefficient, as illustrated in Figure 2. (B/L)

_{best}is the value of B/L at which the best efficiency occurs. The best efficiency ${\epsilon}_{\mathrm{best}}$ is 0.215, and the best (B/L)

_{best}is 0.16 at the lowest K

_{t}= 1000 Pa·m

^{−3}·s. The best efficiency ${\epsilon}_{\mathrm{best}}$ is 0.069, and (B/L)

_{best}is reduced to 0.095, when the turbine coefficient is increased to 36,000 Pa·m

^{−3}·s. By using a trial-and-error method, it was discovered that the turbine coefficient K

_{t}= 3000 Pa·m

^{−3}·s has a maximum ${\epsilon}_{\mathrm{best}}$, that occurs at B/L = 0.16.

_{t}= 3000 Pa·m

^{−3}·s and d/h = 0.25 for the horizontal and vertical motion of the OWC chamber at m = 2 and 3. K

_{t}= 3000 Pa·m

^{−3}·s has the best performance for a fixed OWC as discussed in above.

_{t}= 3000 Pa·m

^{−3}·s. Figure 5a–d show the variations of the amplitude of the oscillatory air volume of the OWC chamber with B/L for the same cases for Figure 4. The non-dimensional amplitude of air pressure is defined as ${p}^{*}=\left({p}_{\mathrm{max}}-{p}_{\mathrm{min}}\right)/\rho g{H}_{\mathrm{i}}$, where ${p}_{\mathrm{max}}$ and ${p}_{\mathrm{min}}$ are the maximum and minimum air pressure in the OWC chamber within one wave period, respectively. The amplitude of the air volume is defined as ${V}^{*}=\left({V}_{\mathrm{max}}-{V}_{\mathrm{min}}\right)/BW{H}_{\mathrm{i}}$, where ${V}_{\mathrm{max}}$ and ${V}_{\mathrm{min}}$ are the maximum and minimum air volume in the OWC chamber within one wave period, respectively. It can be seen from Figure 4a,c that the horizontal motion does not affect the variation of the amplitude of air pressure with B/L much. On the other hand, as shown in Figure 4b,d, the vertical motion of the OWC has a huge impact on the air pressure at the frequency ratios ${R}_{f}$ = 1 and 1.5, and the pressure is increased with an increase of the frequency ratio for the vertical motion.

_{X}/A

_{i}) and vertical motion (A

_{Y}/A

_{i}) of OWCs, where A

_{X,}A

_{Y}and A

_{i}are the oscillatory amplitudes in the x- and y-directions, respectively, and A

_{i}= H

_{i}/2 is the incoming wave amplitude. It is shown that at the highest frequency ratio (${R}_{f}$ = 10), the non-dimensional oscillatory amplitude is almost zero. For both the horizontal and vertical OWC oscillation, the non-dimensional oscillatory amplitude increases with the reduction of ${R}_{f}$. Figure 7a–d show the variation of the non-dimensional water surface elevation at the centre of the OWC chamber for both the horizonal and vertical oscillation. The non-dimensional wave surface in the centre of a chamber is defined as ${\eta}^{*}=\frac{{\eta}_{\mathrm{max}}-{\eta}_{\mathrm{min}}}{{H}_{i}}$, where ${\eta}_{\mathrm{max}}$ and ${\eta}_{\mathrm{min}}$ are the maximum and minimum values of the surface elevation of the chamber, respectively. In Figure 7a,c, the frequency ratio ${R}_{f}$ does not have much effect on the surface elevation at the OWC chamber for all frequency ratios (${R}_{f}$) for the horizontal motions of OWCs at two different mass ratios (m). It is noted that the gauge may not be fixed at the OWC chamber, as the OWC moves horizontally. Figure 7b,d show that ${R}_{f}$ has a significant effect on the wave surface elevation for the vertical oscillation of the OWC.

_{t}= 3000 Pa·m

^{−3}·s for the vertical motion of the OWC device. The value of φ is found to continuously increase with the increase of ${R}_{f}$ until ${R}_{f}$ = 10 for the vertical motion of OWC device. Both A

_{Y}and η* increase if ${R}_{f}$ decreases, but the largest amplitudes of oscillation and wave surface elevation at ${R}_{f}$ = 0 produce the smallest power, as shown in Figure 3, because the very small phase $\phi $ difference between them creates a very small change in the OWC volume or air flow rate. With the increase of ${R}_{f}$, the increase in $\phi $ is in favour of power generation, but the decreases in ${A}_{\mathrm{Y}}$ and η* cause the reduction in power generation. The combination effects of $\phi $, ${A}_{\mathrm{Y}}$, and η* make the maximum best efficiency occurring at ${R}_{f}$ = 10, K

_{t}= 3000 Pa·m

^{−3}·s, and d/h = 0.25.

_{t}= 3000 Pa·m

^{−3}·s, ${R}_{f}$ = 1, and B/L = 0.159 at the horizontal motion of the OWC device. The vorticity ${\omega}_{\mathrm{z}}$ is defined as ${\omega}_{\mathrm{z}}=\partial \mathrm{v}/\partial \mathrm{x}-\partial \mathrm{u}/\partial \mathrm{y}.$ Near the bottom of each wall, two vortices with opposing directions are generated by water flowing both into and out of the OWC chamber at the highest velocities in Figure 9. The vortices are found to be in pairs, and pairs near the positive x-side wall are stronger than those near the negative x-side wall. A new pair of vortices forms when each pair of vortices moves away and dissipates. Figure 10 shows the contours of vorticity for the vertical motion of the OWC device under the same parameters for Figure 9, except ${R}_{f}$ = 10. By observation, the vortices in Figure 10 appear to have stronger vortices than that in Figure 9.

## 4. Conclusions

_{i}= 0.04 m, B = 0.18 m, K

_{t}= 3000 Pa·m

^{−3}·s, d/h = 0.25, and two mass ratios (m = 2 and 3). A wide range of B/L values were simulated for each value of the frequency ratio (${R}_{f}$), and the best efficiency and the B/L where the highest efficiency occurs were defined. First, 15 values of K

_{t}in the range of 1000 Pa·m

^{−3}·s to 36,000 Pa·m

^{−3}·s were simulated for a fixed OWC, and it was found that the K

_{t}= 3000 Pa·m

^{−3}·s had the greatest performance.

- It was found that the frequency ratio affects the OWC with the vertical motion much more than that the OWC with the horizontal motion. The maximum efficiencies for the vertical motion and horizontal motion OWCs occur at the largest and smallest frequency ratios, respectively. At m = 2, the maximum hydraulic efficiency of horizontal motion was 0.291, found at B/L = 0.159 and ${R}_{f}$= 1 and that of vertical motion was 0.270, found at B/L = 0.159 and ${R}_{f}$ = 10.
- The strong vertical motion of the water and OWC at small frequency ratios in the vertical motion case does not create a lot of energy, because the phase difference between the water surface motion and the OWC motion is very small, creating a very small relative motion between them.
- A mounting system’s natural frequency must be sufficiently high to provide high efficiency if the OWC oscillates vertically. However, when the OWC oscillates horizontally, the effect of its natural frequency is very weak.
- When water flows in and out of the OWC chamber, a pair of vortices with opposing directions are created near the bottom end of each OWC wall. The vortices for a horizontal motion OWC with the maximum energy occurring at ${R}_{f}$ = 1 are weaker than the vortices for a vertical motion OWC with the maximum energy occurring at ${R}_{f}$ = 10.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of a two-dimensional wave tank developed to simulate the hydrodynamic performance of (

**a**) sway-only floating (

**b**) heave-only floating OWC in a wave flume; (

**c**) computational mesh near the OWC.

**Figure 2.**Effects of the turbine coefficient for fixed OWC device at (d/h = 0.25) on efficiency with B/L for different turbine coefficient.

**Figure 3.**Variation of the floating OWC device efficiency ε with $B/L$ under nine different frequency ratios (${R}_{f}$ ) at d/h= 0.25, K

_{t}= 3000 Pa·m

^{−3}·s.

**Figure 4.**Variation of the amplitude of air pressures oscillatory of the floating OWC chamber with $B/L$ for K

_{t}= 3000 Pa·m

^{−3}·s and various frequency ratios at d/h = 0.25.

**Figure 5.**Variation of the amplitude of oscillatory air volume of the floating OWC chamber with $B/L$ for K

_{t}= 3000 Pa·m

^{−3}·s and various frequency ratios at d/h = 0.25.

**Figure 6.**Variation of the amplitude of the oscillatory horizontal and vertical motion of the floating OWC chamber for K

_{t}= 3000 Pa·m

^{−3}·s and d/h = 0.25.

**Figure 7.**Variation of the amplitude of the waver surface elevation of the floating OWC chamber $B/L$ for K

_{t}= 3000 Pa·m

^{−3}·s and d/h = 0.25.

**Figure 8.**The phase between the floating OWC displacement and surface elevation at the centre of the OWC for K

_{t}= 3000 Pa·m

^{−3}·s and vertical motion at m = 2.

**Figure 9.**Flow near the OWC represented by streamlines and vorticity contours at horizontal motion, m = 2, B/L = 0.159.

**Figure 10.**Flow near the OWC represented by streamlines and vorticity contours at vertical motion, m = 2, B/L = 0.159.

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**MDPI and ACS Style**

Mia, M.R.; Zhao, M.; Wu, H.; Dhamelia, V.; Hu, P.
Hydrodynamic Performance of a Floating Offshore Oscillating Water Column Wave Energy Converter. *J. Mar. Sci. Eng.* **2022**, *10*, 1551.
https://doi.org/10.3390/jmse10101551

**AMA Style**

Mia MR, Zhao M, Wu H, Dhamelia V, Hu P.
Hydrodynamic Performance of a Floating Offshore Oscillating Water Column Wave Energy Converter. *Journal of Marine Science and Engineering*. 2022; 10(10):1551.
https://doi.org/10.3390/jmse10101551

**Chicago/Turabian Style**

Mia, Mohammad Rashed, Ming Zhao, Helen Wu, Vatsal Dhamelia, and Pan Hu.
2022. "Hydrodynamic Performance of a Floating Offshore Oscillating Water Column Wave Energy Converter" *Journal of Marine Science and Engineering* 10, no. 10: 1551.
https://doi.org/10.3390/jmse10101551