Experimental and numerical investigation on wave height and power take–off damping effects on the hydrodynamic performance of an offshore–stationary OWC wave energy converter

Accepted Manuscript Experimental and numerical investigation on wave height and power take–off damping effects on the hydrodynamic performance of an offshore–stationary OWC wave energy converter

Ahmed Elhanafi, Chan Joo Kim PII:

S0960-1481(18)30283-0

DOI:

10.1016/j.renene.2018.02.131

Reference:

RENE 9862

To appear in:

Renewable Energy

Received Date:

23 September 2016

Revised Date:

24 December 2017

Accepted Date:

28 February 2018

Please cite this article as: Ahmed Elhanafi, Chan Joo Kim, Experimental and numerical investigation on wave height and power take–off damping effects on the hydrodynamic performance of an offshore–stationary OWC wave energy converter, Renewable Energy (2018), doi: 10.1016/j.renene.2018.02.131

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ACCEPTED MANUSCRIPT

3

Experimental and numerical investigation on wave height and power take– off damping effects on the hydrodynamic performance of an offshore– stationary OWC wave energy converter

4

Ahmed Elhanafia,b, Chan Joo Kima

1 2

5 6

a) National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, Tasmania 7250, Australia

7 8

b) Department of Naval Architecture and Marine Engineering, Alexandria University, Alexandria, Egypt

9

Abstract

10

Wave energy is a viable source of ocean renewable energy and research is being

11

conducted worldwide. The Oscillating Water Column (OWC) device is recognised

12

internationally as one of the most promising types of ocean Wave Energy Converters (WECs).

13

To effectively utilize ocean waves for harvesting more energy, offshore OWC devices need to

14

be deployed in deep–water where waves are more energetic. Therefore, the present paper

15

experimentally investigated the hydrodynamic performance of a 3D offshore–stationary OWC

16

device subjected to a wide range of regular wave conditions of different periods and heights and

17

nonlinear power take–off (PTO) damping conditions simulated by an orifice. The experimental

18

results were also employed to validate a 3D incompressible Computational Fluid Dynamics

19

(CFD) model based on the RANS–VOF approach. It was found that the device capture width

20

ratio decreased as wave height increased, especially for wave frequencies higher than device

21

resonance frequency. However, for low–frequency waves under small PTO damping, there was

22

a noticeable improvement in the device capture width ratio. More importantly, results of this

23

study revealed that even with the changes in the device capture width ratio as wave height

24

doubled, the OWC device could extract more wave energy throughout the whole frequency range

25

tested by a maximum of about 7.7 times, particularly for long waves under small PTO damping.

26

Furthermore, the numerical results from the 3D CFD model were in good agreement with the

27

experiments, while the 2D model provided misleading (overestimating) results for high–

28

frequency waves. Keywords: Wave energy; offshore oscillating water column device; OWC; 3D experiments

29 30

and CFD; PTO damping effect; wave height effect. *

Corresponding author. Tel.: +61470352926 E-mail addresses: [email protected] (A. Elhanafi) 1

ACCEPTED MANUSCRIPT 1. Introduction

31 32

To cope with the increasing costs of fossil fuels as well as the environmental impacts due

33

to the excessive use of these fuels, eco–friendly power from renewable energy sources has been

34

under the spotlight and is believed to have an important role in resolving these effects. Ocean

35

wave energy is considered to be one of the most promising renewable energy sources as it can

36

provide noteworthy advantages when compared to other sources such as solar, wind and tidal.

37

Wave energy offers the highest energy density and relatively small potential energy losses when

38

travelling over long distance [1]. A key feature is that Wave Energy Converter (WEC) devices

39

are able to produce power up to ninety percent of the time, whereas other types of renewable

40

energy conversion devices such as solar energy converters and wind turbines can only generate

41

power about twenty to thirty percent of the time [1]. The World Energy Council Report on World

42

Energy Sources [2] estimated the total theoretical wave energy potential of 32,000 TWh/year.

43

This value decreased to 16,000–18,500 TWh/year when the direction of the wave energy and the

44

world coastline alignment were taken into account [3]; however, this is still comparable to the

45

global electricity consumption (energy demand) of about 20,500 TWh in year 2015 [4]. Utilizing

46

these advantages, several techniques have been proposed and developed to extract wave energy.

47

These technologies can be categorised by deployment location (shoreline, nearshore and

48

offshore), type (attenuator, point absorber and terminator) and mode of operation (submerged

49

pressure differential, oscillating wave surge converter, oscillating water column and overtopping

50

device) [1]. The wave energy electricity production by the current technologies and designs of

51

devices when fully mature has been estimated to be 140–750 TWh/year; however, this figure

52

could be increased to 2,000 TWh/year if all potential improvements to these devices are realised

53

[5].

54

The Oscillating Water Column (OWC) device is a WEC that is based on wave–to–

55

pneumatic energy conversion by utilizing ocean waves to drive the motion of a water column

56

inside a pneumatic chamber that has an underwater opening to the ocean. The water column

57

fluctuation (oscillation) compels the air inside being compressed and decompressed which

58

creates a bidirectional airflow. The pneumatic energy is then converted into mechanical energy

59

via forcing the air to flow through a turbine or a power take–off (PTO) system that is connected

60

to a generator to transform the mechanical energy into electricity. Each phase of the energy

61

conversion chain has its own importance in terms of overall performance. The focus of the

62

present study is to investigate the wave–to–pneumatic energy conversion stage. OWC devices

2

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can be installed at different locations (shoreline, nearshore, and offshore) as either fixed or

64

floating structures [6].

65

Investigating the hydrodynamic performance of OWC devices has been extensively

66

performed by different means including analytical, experimental, numerical or in a combination

67

of the aforementioned. Most if not all analytical studies such as McCormick [7], Falcão and

68

Sarmento [8], Evans [9] and Falnes and McIver [10] are based on linear wave theory to model

69

simple OWC device geometries like a thin–walled vertical tube and two parallel vertical thin

70

walls. For complex geometries, the wave–OWC device interactions require numerical modelling

71

that is usually solved using the Boundary Element Method (BEM) [11]. However, these methods

72

are still based on the potential flow assumptions and cannot handle problems that require

73

capturing detailed physics such as strong nonlinearity, complex viscous effects, turbulence and

74

vortex shedding. For extensive reviews of utilizing analytical and potential flow models to study

75

OWC devices, the reader is referred to Refs. [11, 12]. On the other hand, Numerical Wave Tanks

76

(NWTs) based on the Reynolds–averaged Navier–Stokes (RANS) equations do not have the

77

above–mentioned shortcomings. Accordingly, these advantages together with the increasing

78

availability of computational power have made Computational Fluid Dynamics (CFD) more

79

attractive and viable for researchers to study the hydrodynamic performance of OWC devices

80

[13-17]. However, CFD modelling of OWC devices has generally been restricted to 2D

81

geometries and regular waves, and it still requires validation against physical model testing [11].

82

Over the past few decades, many physical scale model experiments have been performed

83

to investigate the effects of different design parameters on the hydrodynamic performance of

84

onshore OWC devices [18-22]. In comparison to the typical onshore and nearshore types of

85

OWC devices, offshore OWC devices allow ocean waves to pass around and underneath the

86

device, which in turn changes the hydrodynamic interactions and the energy balance of such

87

devices [23]. However, there is a very limited number of experiments and/or numerical studies

88

on offshore OWC devices. For example, Iturrioz et al. [24, 25] and Simonetti et al. [26, 27]

89

developed open source CFD codes (IHFOAM) and (OpenFOAM) to model and investigate the

90

hydrodynamic performance of a fixed detached OWC model and a three chamber OWC device,

91

respectively after utilizing wave flume measurements to validate their numerical models.

92

Elhanafi, et al. [23] employed a 2D CFD model using a commercial code (STAR–CCM+) to

93

study the relevance of incident wave amplitude, wave frequency and turbine damping to the

94

energy balance of an offshore stationary OWC device. Recently, He, et al. [28] experimentally

95

investigated the importance of energy extraction and vortex–induced energy loss with respect to 3

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energy dissipation for a pile–supported OWC model embedded breakwater. They discovered that

97

the energy extraction and vortex–induced energy loss are closely related, and the pneumatic

98

damping has a greater influence on the vortex–induced energy loss than the energy extraction.

99

Although He, et al. [28] achieved significant findings in their experimental investigation, it is

100

worth noticing that the experiments were conducted in a 2D wave flume, which constrains wave

101

diffraction and transmission around the device. In addition, the OWC model was subjected to a

102

single small wave height representing weakly nonlinear waves of steepness 0.0123–0.0287, and

103

only intermediate water depth conditions were tested based on the water depth and wave periods

104

provided.

105

To complement previous studies, the main objective of this study is to experimentally

106

investigate the hydrodynamic performance of a 3D offshore OWC device under the action of

107

regular waves of different periods and heights including strong nonlinear waves in deep– and

108

intermediate–water depth conditions. In addition, the effect of changing the PTO damping on

109

device performance is studied. Finally, the capability of a 3D incompressible CFD model based

110

RANS–VOF approach in predicting the device performance is discussed. Both experiments and

111

CFD simulations were performed at a small scale model, and full–scale system dynamics inside

112

the OWC chamber due to air compressibility effects [29] were not modeled in the present study.

113

2. Experimental set–up

114

2.1.

Testing facility

115

The experimental campaigns were performed in the towing tank of the Australian

116

Maritime College (AMC), University of Tasmania, Australia. The tank characteristics are 100 m

117

long, 3.5 m width and a maximum water depth of 1.5 m. Waves are generated by means of a

118

flap–type wavemaker installed at one end, whereas a wave–absorption beach is located at the

119

other end.

120

2.2.

Offshore OWC model

121

The 1:50 scale model offshore OWC device used in these experiments was manufactured

122

of plywood with dimensions illustrated in Fig. 1–a, and it was positioned 15 m from the

123

wavemaker (see Fig. 2). The model was scaled based on Froude’s similitude law such that the

124

water depth (1.5 m) of the towing tank represented 75 m at full–scale. The OWC model had an

125

overall draught of 350 mm and a rectangular chamber (300 mm x 200 mm) with symmetric front

126

and rear lips of 12 mm thickness and 200 mm submergence. Considering that the ratio of the

127

tank width (W) to the model breadth (B) was 8.75 (i.e., W/B > 5), the tank sidewall effects could 4

ACCEPTED MANUSCRIPT 128

be ignored [30]. The columns and pontoons attached to the chamber provide the required

129

buoyancy to permit future experiments over a wider range of wave conditions on the same model,

130

but as a floating OWC device which is tension moored using a similar concept to that proposed

131

in Lye, et al. [31].

132

The chamber top plate of 12 mm thick included a central opening to utilize testing

133

different PTO damping conditions. To simulate the pneumatic damping induced by the PTO

134

system during experiments of small scale models, it is common to use either an orifice (or a slot)

135

to simulate an impulse turbine (nonlinear/quadratic pressure–flow rate characteristics) or a

136

porous medium to mimic a Wells turbine (linear characteristics) [18, 29]. In the present study,

137

an orifice of different diameters shown in Fig. 1–b was manufactured via a 3D printer and used

138

to model nonlinear PTO damping. Each orifice was defined by its radius and opening ratio (the

139

ratio between the orifice area and the chamber waterplane area (a x b) in percentage, %).

140

Considering the 12 mm thickness of the chamber top plate and the orifice diameter, orifices R2–

141

R5 were classified as thin–walled openings (i.e., the ratio between orifice thickness (12 mm) and

142

orifice diameter is less than 0.5), while orifice R1 was classified as a thick–walled opening [32].

143 144 145

Fig. 1. 1:50 scale model of an offshore–stationary OWC device. (a): main dimensions and (b): orifice diameters

5

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146 147 148 149

Fig. 2. Photographs showing the AMC towing tank with the OWC model, looking towards the beach (left) and test rig in–situ (centre: front view and right: back–side view) 2.3.

Instrumentation and test conditions

150

In the experiments (see Fig.3 for experiment layout), three resistance–type wave gauges

151

were used to measure the instantaneous free surface elevation in the tank and inside the OWC

152

chamber. One wave gauge (G1) was positioned 6 m from the wavemaker to measure the free

153

surface elevation of the incident wave. For rectangular shaped OWC devices like the model

154

considered in this study, measurements of free surface motion inside the OWC chamber are

155

usually carried out using one [33], two [28] or three [21] wave gauges. Herein, two wave gauges

156

(G2 and G3) were situated inside the OWC chamber to spatially average the free surface

157

elevation (ƞOWC). To measure the differential air pressure between the OWC pneumatic chamber

158

and exterior atmosphere, two pressure sensors (Honeywell–TruStability–001PD TSC Series)

159

marked as P1 and P2 were installed on the top surface of the OWC chamber to average the

160

measured pressure (∆𝑝(𝑡)). All measurements were sampled at 200Hz. During data processing,

161

a low pass digital filter (Butterworth–filter [34]) with a 5 Hz cut–off frequency and 10th order

162

was applied using the Matlab function filtfilt (non–causal, zero–phase filter) [35].

6

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163

Fig. 3. Experiment layout (not to scale)

164 165

The physical measurements were designed to study the effect of incoming wave period,

166

wave height and PTO damping on the hydrodynamic performance of the offshore OWC model

167

described in Section 2.2. The experiments systematically investigated the following variables:

168

two regular wave heights H = 50 and 100 mm, ten wave periods T = 0.9, 1.0, 1.1, 1.2, 1.3, 1.4,

169

1.5, 1.6, 1.8, 2.0 s and six orifice diameters: R0 – R5 (see Fig. 1–b). Within the range of wave

170

conditions tested, wave steepness varied between 0.009 – 0.080 and both intermediate (T = 1.4–

171

2.0 s) and deep–water (T = 0.9–1.3 s) conditions were covered.

172

2.4.

Data analysis and OWC device performance

173

The capture width ratio (ζ) [33, 36] or the dimensionless capture width [37] that best

174

represents the hydrodynamic performance of a WEC is defined as in Eq. (1) for the OWC device

175

considered in this study. 7

ACCEPTED MANUSCRIPT 176

ζ = 𝑃𝐸/(𝑃𝐼𝑎)

177

where 𝑃𝐼 is the mean incident wave power (energy flux) per unit width, which is defined as the

178

product of the wave energy (𝐸𝐼) (potential and kinetic) per unit ocean surface area and the

179

group velocity (𝐶𝑔) [38] as given in Eqs. (2) and (3), while 𝑃𝐸 is the time–averaged extracted

180

pneumatic power determined as in Eq. (4) [23, 39, 40] and a is chamber width (Fig. 1–a).

181

1 2 𝑃𝐼 = 𝜌𝑔𝐴 𝐶𝑔 2

182

𝜔 2𝑘ℎ 𝐿 𝐿 1+ for intermediate ‒ water conditions <ℎ< 2𝑘 sinh(2𝑘ℎ) 20 2 𝐶𝑔 = 1 𝐿 𝐿 for deep ‒ water conditions ℎ > 2𝑇 2

183

where 𝐴 is incident wave amplitude (defined as half the measured wave height, H), 𝜔 is wave

184

angular frequency, 𝑘 is wave number given by the dispersion relationship

185

incoming wavelength, T is wave period, h is still water depth, 𝜌 is water density and 𝑔 is

186

gravitational acceleration.

187

𝑃𝐸 =

{

(

(1)

(2)

)

(

)

( )

(3)

2

1 𝑇

𝜔 𝑔

= 𝑘tanh(𝑘ℎ), L is

𝑇

∫ ∆𝑝(𝑡)𝑞(𝑡) 𝑑𝑡

(4)

0

188

where 𝑞(𝑡) is airflow rate through the orifice, which can be measured by two commonly accepted

189

approaches: (1) using a pre–calibrated orifice together with the pressure measurement [41] and

190

(2) by measuring the free surface elevation inside the OWC chamber [21, 28, 33, 40, 41] with

191

incompressible flow assumption. The second approach was utilized in the present experiments

192

as follows. Using the averaged chamber free surface oscillations (ƞOWC), the free surface vertical

193

velocity (V) was calculated by differentiating the measured time–series data (i.e., V(t) =

194

dƞOWC/dt). Having defined the chamber free surface vertical velocity and assumed

195

incompressible air for the small model–scale used in these experiments [42], airflow rate (𝑞(𝑡))

196

was calculated by Eq. (5). It is worth highlighting that air compressibility effects must be

197

considered when scaling–up the results presented in this paper to assess device performance at

198

full–scale.

199

𝑞(𝑡) = 𝑉(𝑡) 𝑏𝑎

(5) 8

ACCEPTED MANUSCRIPT 200 201

where b and a are the chamber length and width, respectively (see Fig. 1–a). 3. Results and discussion

202

The hydrodynamic performance of the OWC device described in Section 2 was initially

203

assessed when subjected to ten different wave periods and two wave heights H = 50 mm and 100

204

mm under three PTO damping conditions (R2, R4 and R5). Fig. 4 illustrates the relevance of these

205

variables to the amplification factor (ƞmax/A, defined as the ratio between the maximum free

206

surface oscillation inside the chamber ƞmax and the incoming wave amplitude A), the pressure

207

coefficient (CP) given by Eq. (6) and the capture width ratio (ζ). In this study, a dimensionless

208

parameter 𝐾𝑏 where 𝐾 = 𝜔 /𝑔 was used to refer to the changes in wave period/frequency [23].

209 210 211

2

𝐶𝑃 =

∆𝑝max

(6)

𝜌𝑔𝐴

where ∆𝑝max is the differential air pressure amplitude defined as in Eq. (7) ∆𝑝max =

[

∆𝑝max (crest value,

+ ve) ‒ ∆𝑝min (trough value, ‒ ve)

2

]

(7) avg ‒ 5 cycles

212

Overall, both wave heights provided almost the same general trends for all assessed

213

parameters as illustrated in Fig. 4. As wave frequency increased (Kb increased), the amplification

214

factor (Fig. 4–a) gradually increased until peaked at a certain Kb value that increased as PTO

215

damping decreased. Following this peak, a steep drop in ƞmax/A was observed as Kb further

216

increased. This was in agreement with what has recently been reported experimentally for land–

217

based OWC devices [21] and numerically with 2D CFD for offshore OWC devices [23]. Similar

218

results were also found for the larger wave height, H = 100 mm (Fig. 4–d), except the peak value

219

that was shifted to a lower frequency. This was more pronounced under PTO damping cases of

220

R2 and R4 where maximum ƞmax/A was achieved under the longest wave tested of Kb = 0.30.

221

Although the incident wave height increased twofold (i.e., H = 100 mm), the resulted

222

amplification factors were smaller than those for H = 50 mm over the entire frequency range,

223

which revealed a nonlinear relationship between chamber maximum free surface oscillation and

224

incoming wave height. This could be attributed to the increase in chamber free surface

225

deformation and water sloshing as wave height increased [23].

226

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227 228 229 230

Fig. 4. Impact of wave height and frequency on the amplification factor (ƞmax/A), pressure coefficient (CP) and device capture width ratio (ζ) under three damping conditions. (a), (b) and (c) for H = 50 mm and (d), (e) and (f) for H = 100 mm

231

The pressure coefficient (CP) (Figs. 4–b and 4–e) increased as Kb increased, reached a

232

maximum value at a certain frequency (Kb) representing (or close to) the chamber resonance

233

frequency, and then declined with a further increase in Kb (i.e., wavelength shortened). The

234

device resonance frequency was shifted to a lower value as PTO damping increased. This effect

235

could obviously be seen in the case of the closed orifice (full–damping, R0) where a maximum

236

pressure coefficient of CP = 0.68 for both H = 50 mm and H = 100 mm was found under the

237

longest wave tested (Kb = 0.30). This was also observed by Iturrioz, et al. [25] and He and Huang

238

[32] for similar offshore OWC devices that have been tested in 2D wave flumes under a constant

239

wave height.

240

Although Iturrioz, et al. [25] did not present results for the amplification factor, it was

241

stated that the maximum pressure happens close to the resonance period of the chamber, and at

242

that period the chamber maximum free surface oscillation also occurs. Conversely, Simonetti, et

243

al. [26], Ning, et al. [21] and Elhanafi, et al. [23] found that the maximum pressure amplitude

244

occurs at a frequency higher than that of the chamber maximum free surface oscillation for a

245

given wave height, which is in line with the results presented in Figs. 4–a and 4–b of this study.

246

However, Figs. 4–d and 4–e show that under the lowest damping tested (orifice R5) the maximum

247

pressure coefficient and amplification factor occurred nearly at the same frequency (Kb = 0.62),

248

which supports the findings of Iturrioz, et al. [25]. This contradiction can be explained by

249

considering the fact that the chamber differential air pressure is related to the chamber free

250

surface oscillation rate (slope) not to the oscillation amplitude solely [23]. For illustration, Fig.

10

ACCEPTED MANUSCRIPT 251

5 shows the instantaneous variations (time–series results) of chamber free surface oscillation

252

(ƞOWC), chamber differential air pressure (∆𝑝), airflow rate (𝑞) and extracted pneumatic power (

253

𝑃𝐸) for a given PTO damping induced by orifice R4 under two wave heights H = 50 mm (Fig.

254

5–left) and H = 100 mm (Fig. 5–right) at different Kb values. Although low–frequency waves of

255

Kb = 0.47 at H = 50 mm (Fig. 5–a) and Kb = 0.37 at H = 100 mm (Fig. 5–d) provided higher

256

chamber free surface oscillation amplitudes (i.e., higher amplification factors considering a

257

constant wave height of H = 50 or 100 mm), larger pressure amplitudes were found at a higher

258

frequency of Kb = 0.84 for H = 50 mm (Fig. 5–b) and Kb = 0.62 for H = 100 mm (Fig. 5–e).

259

This is because at these frequencies (i.e., Kb = 0.84 and 0.62), the smaller chamber free surface

260

oscillation amplitudes were accompanied by shorter wave periods, which in turn resulted in a

261

higher free surface oscillation rate inside the chamber (i.e., a higher slope with respect to time–

262

axis) that are demonstrated by the higher airflow rate and pressure amplitudes shown in Figs.

263

5–b and 5–e. Except the closed orifice (R0), increasing the wave height (Fig. 4–e) resulted in: (1)

264

higher pressure coefficients throughout the whole Kb range tested, which further explains the

265

associated reduction in the amplification factors and (2) a slight reduction in device resonance

266

frequency that could be due to the coarse resolution of the Kb range tested (i.e., large wave period

267

interval of 0.1 s, see Section 2.3).

268 269 270 271

Fig. 5. Effects of wave frequency and wave height on the instantaneous chamber free surface oscillation (ƞOWC), chamber differential air pressure (∆𝑝), airflow rate (𝑞) and extracted pneumatic power (𝑃𝐸) at a constant PTO damping (R4)

272

The capture width ratio of the OWC model is illustrated in Figs. 4–c and 4–f for H = 50

273

mm and 100 mm, respectively. To explain the changes in device capture width ratio, it is 11

ACCEPTED MANUSCRIPT 274

important to note that both the differential air pressure and the airflow rate are the governing

275

parameters for the extracted pneumatic power as given in Eq. (4). Although the results of the

276

airflow rate through the orifice are not presented in Fig. 4, it is expected the airflow rate to follow

277

the same trend of the chamber differential air pressure at a given PTO damping [23]. This is

278

because both parameters are related by a damping coefficient (𝐶) as given in Eq. (8) for nonlinear

279

PTO system [14] that was simulated by an orifice in the present study. 1 2

∆𝑝 𝑞

280

𝐶=

281

This correlation can also be seen in the results presented in Fig. 5. As a result, the

282

extracted pneumatic power is also expected to have the same trend of the chamber differential

283

air pressure (see Fig. 5) such that the maximum extracted pneumatic power was achieved at the

284

same frequency of the maximum pressure. On the other side, as Kb increased the group velocity

285

of the incoming wave decreased as given by Eq. (3), which in turn reduced the available

286

incoming wave power as given by Eq. (2). Therefore, the device capture width ratio gradually

287

increased according to Eq. (1) as Kb increased until peaked at a certain frequency of resonance

288

(for example, at Kb = 0.71 and 0.62 for H = 50 mm and 100 mm respectively under a PTO

289

damping of orifice R4). Following that peak, the pneumatic power decreased due to the drop in

290

the pressure and the airflow rate (see Figs. 5–c and 5–f); hence, the capture width ratio (ζ)

291

decreased with a further increased in Kb (see Figs. 4–c and 4–f). For instance, considering the

292

condition of H = 50 mm and orifice R4 (Fig. 4–c), the capture width ratio (ζ) improved from 0.04

293

at Kb = 0.30 to a maximum of about 0.25 at Kb = 0.71, then dropped to 0.09 at Kb = 1.0 and

294

diminished at the highest frequency tested of Kb = 1.5. The reason for the low capture width

295

ratio at the high–frequency zone could be that, the frequency response of the system is far from

296

the resonance condition, which is demonstrated by the small amplitudes of the chamber free

297

surface oscillation, airflow rate and differential air pressure shown in Fig. 5–c for Kb = 1.21 and

298

H = 50 mm.

(8)

299

Although the maximum capture width ratio reported in this paper (ζmax = 0.26 for a PTO

300

damping induced by orifices R2 and R3) is quite smaller than what was experimentally found for

301

typical onshore OWC devices [21, 33], it is worth noting that for a two–dimensional chamber

302

with a vertical plane of symmetry (i.e., identical front and rear lips draught), Sarmento [18]

303

predicted using linear theory [9] a maximum value of 0.5 for the ratio between the energy

304

absorbed by the OWC structure/chamber (i.e., the rest of the incident wave energy that was not 12

ACCEPTED MANUSCRIPT 305

reflected and/or transmitted on the lee side of the chamber, which represents the maximum

306

available energy to be extracted for electricity generation [23]) and the incident wave energy.

307

However, a maximum capture width ratio of 0.5 was not achievable due to energy losses in terms

308

of viscous, turbulence and vortex losses at chamber lips and PTO system [18, 23, 28]. Using 2D

309

wave flume experiments, He, et al. [28] reported a maximum capture width ratio of 0.25–0.37

310

(for different symmetrical lip draughts and PTO damping conditions, see Fig. 4 in He, et al. [28]),

311

which is slightly higher than the maximum capture width ratio presented in this paper. This

312

difference could be related to the 2D behaviour of He et al.’s experiments that will be briefly

313

discussed in Section 4.3 as well as to the different lip draughts tested. The relevance of chamber

314

underwater geometry to the capture width ratio of offshore OWC devices has been found by

315

Elhanafi, et al. [43] to be significant. They tested several symmetrical and asymmetrical lips

316

draught and thickness of a 2D OWC chamber with length (b) similar to that of the OWC model

317

tested in this study (i.e., b = 300 mm, see Fig. 1–a), and reported a significant improvement in

318

the capture width ratio up to 0.79, which is close to that of typical onshore OWC devices. This

319

emphasises the importance of investigating the capability of CFD modelling (Section 4) in

320

replicating the 3D physical measurements provided in this study so that it can be used for

321

optimizing the underwater geometry of offshore OWC devices for a maximum capture width

322

ratio.

323

Generally, Figs. 4–c and 4–f show that as wave height increased the maximum capture

324

width ratio decreased. For instance, the maximum capture width ratio decreased from 0.26 for

325

H = 50 mm at Kb = 0.62 and orifice R2 to 0.23 for H = 100 mm at the same frequency (Kb =

326

0.62) and PTO damping. These results are in agreement with the CFD results of Luo, et al. [15]

327

and Kamath, et al. [44] who found that the capture width ratio of onshore OWC devices

328

decreased as wave height increased. Ning, et al. [21] concluded a negligible effect of wave height

329

on device resonance under a PTO damping induced by an orifice with 0.66 % opening ratio. In

330

the present study, similar results were observed only for the conditions with high PTO damping

331

such as R1 (0.5 %, not presented in Fig. 4) and R2 (1.0 %), but for all other damping conditions

332

(i.e., R > R2), there was a slight reduction in the resonance frequency as wave height increased,

333

which again could be attributed to the large wave period interval (0.1 s) used in this study. The

334

effect of increasing the wave height on the device capture width ratio was influenced by the wave

335

frequency and the PTO damping such that under high steepness waves (strongly nonlinear waves

336

of high Kb) increasing the wave height decreased the capture width ratio, while at low–frequency

337

waves (weakly nonlinear waves of low Kb), there was a noticeable improvement in device

338

capture width ratio as wave height increased, especially for low and intermediate PTO damping 13

ACCEPTED MANUSCRIPT 339

conditions. Similar observations were previously reported for 2D onshore and offshore OWC

340

devices using numerical simulations [23, 45] and physical experiments [20, 21].

341

Previous experimental research on onshore OWC devices [20, 21] classified the

342

pneumatic damping exerted by the turbine as the factor that most affects the device capture width

343

ratio. Accordingly, the influence of the six orifices tested (see Fig. 1–b) on the amplification

344

factor, the pressure coefficient and the capture width ratio is demonstrated by contour plots in

345

Fig. 6 for all wave heights and Kb values tested in these experiments.

346 347 348

Fig. 6. PTO damping effects on the amplification factor (ƞmax/A), pressure coefficient (CP) and device capture width ratio (ζ) for different wave conditions

349

Figs. 6–a and 6–d illustrate that the amplification factor for both wave heights increased

350

as the opening ratio increased (i.e., PTO damping decreased). This in turn increased the free

351

surface oscillation rate for each wave period (Kb), and hence the airflow rate also increased,

352

especially for Kb < 1.0 under intermediate and large opening ratios (for example, see time–series

353

results for Kb = 0.71, H = 50 mm in Fig. 7–left and Kb = 0.62, H = 100 mm in Fig. 7–right).

354

Opposite trends were found for the chamber differential air pressure (Figs. 6–b, 6–e and Fig. 7).

355

These contrary effects of the PTO damping on the variables (∆𝑝 and 𝑞) that control the extracted

356

pneumatic power highlight the importance of selecting an optimum PTO damping to tune the

357

device to a given frequency (Kb) range for maximizing the extracted power (see Fig. 7) and the

358

device capture width ratio (see Figs. 6–c and 6–f). For each wave condition (period and height),

359

there was an optimum opening ratio (damping) that provided a maximum capture width ratio.

360

Reducing the PTO damping (increasing the opening ratio) shifted the maximum capture width

361

ratio to a higher frequency (Kb), but this effect was less pronounced for the larger wave height

14

ACCEPTED MANUSCRIPT 362

of H = 100 mm. Under the smaller wave height H = 50 mm, the device could efficiently extract

363

wave energy over a broader frequency bandwidth (0.54 < Kb < 0.84) and opening ratios. For

364

instance, Fig. 7–left shows that the device extracted almost the same power for a wave condition

365

of H = 50 mm and Kb = 0.71 when applying different PTO damping conditions corresponding

366

to opening ratios of 2 % (Fig. 7–b) and 3 % (Fig. 7–a). On the other hand, increasing the wave

367

height reduced these frequency and opening ratio ranges, but under small PTO damping

368

conditions (opening ratio > 1.0 %), the large wave height improved the device capture width

369

ratio at the low–frequency zone (Kb < 0.62).

370

373

Fig. 7. Impact of PTO damping on the instantaneous chamber free surface oscillation (ƞOWC), chamber differential air pressure (∆𝑝), airflow rate (𝑞) and extracted pneumatic power (𝑃𝐸) at a two wave conditions. Left: H = 50 mm, Kb = 0.71 and right: H = 100 mm, Kb = 0.62

374

Elhanafi, et al. [23] concluded that increasing the wave height not only increases the

375

energy losses, but also the extracted pneumatic energy. Herein, regardless of the variations in

376

the device capture width ratio due to changing the wave height, it was found that the extracted

377

pneumatic energy given by Eq. (9) increased by 2.5–7.7 times as wave height doubled from H =

378

50 mm to 100 mm (i.e., the incident wave energy increased four times), which is demonstrated

379

by the contour plot in Fig. 8. This effect was found to increase as wavelength increased (i.e., Kb

380

decreased) and PTO damping decreased. This larger extracted energy highlights the

381

effectiveness of deploying such offshore OWC devices in deep–water locations where waves are

382

more energetic.

383

𝐸𝐸 = 𝑃𝐸 𝑇

371 372

(9) 15

ACCEPTED MANUSCRIPT

384 385 386 387

Fig. 8. Effect of increasing the wave height from H = 50 mm (H50) to H = 100 mm (H100) on the extracted pneumatic energy 4. CFD modelling and validation

388

As mentioned in the introduction (Section 1), CFD modelling became a viable tool that

389

has previously been employed to investigate the hydrodynamic performance of 2D OWC

390

devices. However, apart from being computationally expensive, CFD requires experimental

391

validation [11]. Therefore, the above–described physical measurements are utilized in this

392

section to validate the developed 3D CFD model. All numerical computations were performed

393

using the RANS–VOF solver in the commercial CFD code STAR–CCM+ that utilizes a finite

394

volume method and the integral formulation of the conservation equations for mass and

395

momentum, while a predictor–corrector approach is used to link the continuity and momentum

396

equations [46].

397

4.1.

Governing equations and numerical settings

398

The CFD model developed in this section assumes incompressible flow since it has been

399

found that air compressibility is negligible in small model–scale OWC devices such that tested

400

in this study [25, 42]. The CFD model solves the continuity and RANS equations to describe the

401

flow motion of the incompressible fluid. To solve the RANS equations, the instantaneous

402

velocity and pressure fields in the Navier–Stokes equations are decomposed into mean value and

403

fluctuating components, and then time–averaged. Consequently, additional unknowns called

404

Reynolds stresses are introduced. To mathematically close the equations, a turbulence model

405

must be utilized to relate Reynolds stresses to the mean flow variables. In this paper, the two–

406

equation shear stress transport (SST) 𝑘–𝜔 turbulence model was implemented since it has been

407

found that 𝑘–𝜔 (SST) provides superior results to k–ɛ in capturing detailed flow field (velocity

16

ACCEPTED MANUSCRIPT 408

and vorticity) and estimating the energy losses in OWC devices when compared to experiments

409

[16].

410

To model and track the free surface motion, the Volume of Fluid (VOF) method

411

introduced by Hirt and Nichols [47] was used. The numerical settings of the CFD model were as

412

follows. Time modelling: implicit unsteady with second–order temporal discretization scheme,

413

flow modelling: segregated flow model with second–order upwind convection scheme (the

414

segregated flow solver controls the solution update for the segregated flow model according to

415

the SIMPLE algorithm) and multiphase model: VOF with segregated VOF solver that controls

416

the solution update for the phase volume fractions (i.e. solve the discretized volume–fraction

417

conservation equation for each phase present in the flow) with second–order convection scheme

418

[46].

419

4.2.

Computational domain, mesh, boundary and initial conditions

420

Ocean waves are the primary exciting source acting on offshore structures such as OWC

421

devices; therefore, accurate modelling of these waves is crucial for providing a good estimation

422

of the hydrodynamic loads and predicting the structure response and performance [48]. Referring

423

to Fig. 9–a, the computational fluid domain of this study had an overall length (in x–direction)

424

of 10L (L is wavelength) including 1L for the wave–damping zone at the end of the NWT, and

425

the OWC model was positioned at a distance of 5L from the wave inlet boundary. This set–up

426

was proven to be appropriate for capturing eight wave cycles up to a distance of 8L from inlet

427

boundary without being influenced by waves reflected from the outlet boundary assigned at the

428

end of the wave–damping zone [23]. To numerically mimic the width of AMC’s physical towing

429

tank without increasing the computation cost and time, the width of the NWT was set as half the

430

width of the towing tank (i.e., W/2 = 1750 mm) with a symmetry plane. The assumption of using

431

a symmetry plane and its impact on the numerical results was found to be negligible [42]. The

432

domain height was fixed at 2500 mm including 1500 mm (still water depth) for the water phase

433

and 1000 mm for the air phase. The tank side and bottom were defined as slip walls, while

434

hydrostatic wave pressure was defined at the top and outlet boundaries.

435

Mesh generation was performed using the automatic meshing technique in STAR–CCM+

436

with a trimmed cell mesher. A base cell size of 400 mm was applied to the whole computational

437

domain with progressive refinements at the free surface zone (the height of the free surface zone

438

was set as 1.5 times the wave height) with at least 16 and 75 cells per wave height (z–direction)

439

and wavelength (x–direction) respectively, which were close to what is recommended by the

440

ITTC [49] and CD–Adapco [46]. The cell aspect ratio (∆x/∆z, the ratio between cell size in x 17

ACCEPTED MANUSCRIPT 441

and z directions) at the free surface zone was set to be not more than 16 as recommended by

442

Elhanafi, et al. [23]. In the tank transverse (y) direction, a cell size of ∆y = 100 mm was used as

443

recommended by Elhanafi, et al. [42] among the three different sizes tested. The time step (∆t)

444

was carefully selected for each wave period (T) so that it satisfied the time step requirement (∆t

445

= T/2.4n, where n is number of cells per wavelength) by CD–Adapco [46] with a second–order

446

temporal discretization scheme. This resulted in a Courant number (C = Cg.∆t/∆x, where Cg is

447

the wave group velocity) [46] smaller than 0.5; hence, the High–Resolution Interface Capturing

448

(HRIC) scheme that is suited for tracking sharp interfaces [46] was guaranteed to be used during

449

the simulations. The OWC non–slip surfaces were refined with 6.25 mm and 0.78125 mm cell

450

surface size for the OWC walls and the PTO orifice opening, respectively as shown in Fig 9–b.

451

For all non–slip walls (OWC and PTO), a first cell height equivalent to y + ≅1 was used with a

452

growth rate of 1.5, ten prism layers and all y+ wall treatment. The initial conditions used in the

453

present CFD model were set as follows (see Fig. 9–c). The water level was defined at the desired

454

still water depth of z = h = 1.5 m, waves were prescribed by the velocity components at the wave

455

velocity inlet boundary and were fully generated throughout the whole domain until the front lip

456

of the OWC model by specifying that point on the water level (i.e., x = 5L).

18

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457 458 459 460

Fig. 9. Computational fluid domain (not to scale). (a): 3D mesh with boundary conditions, (b): OWC model surface mesh and (c): initial conditions 4.3.

Comparison between CFD and experimental results

461

To validate the developed 3D CFD model, the capture width ratio curve from the CFD

462

results shown Fig. 10 for a constant wave height H = 50 mm and PTO damping (R3) was

463

compared to the experimental results provided in Section 3. In this figure, 2D CFD results (with

464

the same pneumatic chamber length (b) and a slot opening of the same opening ratio as R3)[43]

465

were included to highlight 3D effects. It can be seen that CFD results from the 3D model were

466

in good agreement with the physical measurements, whereas the 2D model overestimated device

467

capture width ratio for the high–frequency zone. These differences could be attributed to the

468

constraints in the 2D modelling that did not let waves pass around the OWC device, especially

469

for high–frequency (short) waves when wave diffraction was important [50]. This highlights the

470

importance of using 3D CFD models in assessing the performance of offshore OWC devices.

19

ACCEPTED MANUSCRIPT

471 472 473

Fig. 10. Comparison between CFD (3D and 2D[43]) and experiments for device capture width ratio

474

Additional validations in time–series for incident wave elevation (ƞ1), average free

475

surface elevation inside the chamber (ƞOWC) and chamber differential air pressure (∆𝑝) are shown

476

in Fig. 11 for four different conditions. The normalized root mean square deviation (NRMSD)

477

given by Eq. (10) was found to be in the range of 1.9–5.0 %, which further illustrates the good

478

agreement between the 3D CFD results and the physical measurements.

479

1 1 NRMSD = 𝑥max ‒ 𝑥min 𝑁

𝑁

∑ (𝑥 ‒ 𝑦 ) 𝑖

2

(10)

𝑖

𝑖=1

480

where 𝑥𝑖, 𝑦𝑖, are measured and estimated values form the experiment and the 3D CFD model,

481

respectively, 𝑁 is number of data point (here, data from five wave cycles were considered), 𝑥max,

482

𝑥min are maximum and minimum measured values from the experiment, respectively.

483

20

ACCEPTED MANUSCRIPT

484 485 486 487 488 489

Fig. 11. 3D CFD results vs. time–series physical measurements for different wave conditions at a constant PTO damping (R3). (a): H = 50 mm and Kb = 0.84 (T = 1.2 s), (b): H = 50 mm and Kb = 0.62 (T = 1.4 s), (c): H = 50 mm and Kb = 0.47 (T = 1.6 s) and (d): H = 100 mm and Kb = 0.84 (T = 1.2 s) 5. Conclusions

490

The hydrodynamic performance of a 1:50 scale offshore OWC model was experimentally

491

investigated through 120 test conditions including different wave heights, wave periods and PTO

492

damping. In addition, a fully nonlinear 3D incompressible CFD model based on the RANS–VOF

493

approach was developed and validated against the physical measurements. The following main

494

conclusions can be drawn from the present study based on the incompressible flow assumption

495

at a small scale model and cannot be used at full–scale unless air compressibility effects are taken

496

into account.

497

For a given wave height, the maximum free surface elevation inside the chamber did not

498

occur at the resonance frequency of the OWC chamber but at a lower frequency. The relation

21

ACCEPTED MANUSCRIPT 499

between chamber maximum free surface oscillation and incoming wave amplitude was nonlinear

500

such that the amplification factor decreased as wave height increased.

501

The PTO damping provided an important parameter that could be adjusted to tune the

502

OWC device to a given wave frequency range. As wave height increased, the capture width ratio

503

decreased and increased for wave frequencies higher and lower than the device resonance

504

frequency, respectively. However, the differential air pressure as well as the airflow rate

505

increased as wave height increased, which in turn increased the extracted pneumatic power. This

506

proved the possibility of extracting more wave energy under large waves in offshore locations,

507

especially for low–frequency waves under small PTO damping.

508

Utilizing 2D CFD models to investigate the hydrodynamic performance of offshore

509

OWC devices could significantly overestimate the device capture width ratio, especially for the

510

peak value and the high–frequency zone. In contrast, the numerical results from the 3D CFD

511

model were in good agreement with physical measurements including incident wave elevation,

512

chamber free surface elevation, chamber differential air pressure and device capture width ratio.

513

Acknowledgements

514

The authors would like to gratefully acknowledge the technical support during experiments

515

from Associate Professor Gregor MacFarlane, Dr Alan Fleming, Dr Zhi Leong, Mr Tim

516

Lilienthal and Mr Kirk Meyer, Australian Maritime College (AMC), University of Tasmania,

517

Australia. In addition, the authors thank Mr Liam Honeychurch (AMC) for constructing the

518

tested model. The financial support from the National Centre for Maritime Engineering and

519

Hydrodynamics, Australian Maritime College, University of Tasmania, Australia is also

520

acknowledged.

521

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ACCEPTED MANUSCRIPT

Highlights 

The hydrodynamic performance of a 3D offshore OWC device was experimentally tested.



Effects of wave height and wave period and PTO damping were investigated.



Results from 3D CFD modelling were superior to 2D when compared to experiments.

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The extracted pneumatic energy significantly increased as wave height increased.